Step | Hyp | Ref
| Expression |
1 | | peano2 7624 |
. . 3
⊢ (𝑁 ∈ ω → suc 𝑁 ∈
ω) |
2 | | satfvsucsuc.s |
. . . 4
⊢ 𝑆 = (𝑀 Sat 𝐸) |
3 | 2 | satfvsuc 32897 |
. . 3
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ suc 𝑁 ∈ ω) → (𝑆‘suc suc 𝑁) = ((𝑆‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))})) |
4 | 1, 3 | syl3an3 1166 |
. 2
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (𝑆‘suc suc 𝑁) = ((𝑆‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))})) |
5 | | orc 866 |
. . . . . . 7
⊢ (𝑠 ∈ (𝑆‘suc 𝑁) → (𝑠 ∈ (𝑆‘suc 𝑁) ∨ ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))))) |
6 | 5 | a1i 11 |
. . . . . 6
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (𝑠 ∈ (𝑆‘suc 𝑁) → (𝑠 ∈ (𝑆‘suc 𝑁) ∨ ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))))) |
7 | | satfvsucsuc.a |
. . . . . . . . . . . . . . . . 17
⊢ 𝐴 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣))) |
8 | 7 | eqeq2i 2752 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝐴 ↔ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) |
9 | 8 | anbi2i 626 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ↔ (𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣))))) |
10 | 9 | rexbii 3162 |
. . . . . . . . . . . . . 14
⊢
(∃𝑣 ∈
(𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ↔ ∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣))))) |
11 | | satfvsucsuc.b |
. . . . . . . . . . . . . . . . 17
⊢ 𝐵 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)} |
12 | 11 | eqeq2i 2752 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝐵 ↔ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}) |
13 | 12 | anbi2i 626 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 =
∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵) ↔ (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)})) |
14 | 13 | rexbii 3162 |
. . . . . . . . . . . . . 14
⊢
(∃𝑖 ∈
ω (𝑥 =
∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵) ↔ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)})) |
15 | 10, 14 | orbi12i 914 |
. . . . . . . . . . . . 13
⊢
((∃𝑣 ∈
(𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ↔ (∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))) |
16 | 15 | rexbii 3162 |
. . . . . . . . . . . 12
⊢
(∃𝑢 ∈
(𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ↔ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))) |
17 | 16 | bicomi 227 |
. . . . . . . . . . 11
⊢
(∃𝑢 ∈
(𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)})) ↔ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))) |
18 | | 3simpa 1149 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) |
19 | 1 | ancri 553 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈ ω → (suc
𝑁 ∈ ω ∧
𝑁 ∈
ω)) |
20 | 19 | 3ad2ant3 1136 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (suc 𝑁 ∈ ω ∧ 𝑁 ∈
ω)) |
21 | 18, 20 | jca 515 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω))) |
22 | | sssucid 6250 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑁 ⊆ suc 𝑁 |
23 | 22 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → 𝑁 ⊆ suc 𝑁) |
24 | 2 | satfsschain 32900 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω)) → (𝑁 ⊆ suc 𝑁 → (𝑆‘𝑁) ⊆ (𝑆‘suc 𝑁))) |
25 | 24 | imp 410 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω)) ∧ 𝑁 ⊆ suc 𝑁) → (𝑆‘𝑁) ⊆ (𝑆‘suc 𝑁)) |
26 | 21, 23, 25 | syl2an2r 685 |
. . . . . . . . . . . . . . 15
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (𝑆‘𝑁) ⊆ (𝑆‘suc 𝑁)) |
27 | | undif 4372 |
. . . . . . . . . . . . . . 15
⊢ ((𝑆‘𝑁) ⊆ (𝑆‘suc 𝑁) ↔ ((𝑆‘𝑁) ∪ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))) = (𝑆‘suc 𝑁)) |
28 | 26, 27 | sylib 221 |
. . . . . . . . . . . . . 14
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → ((𝑆‘𝑁) ∪ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))) = (𝑆‘suc 𝑁)) |
29 | 28 | eqcomd 2745 |
. . . . . . . . . . . . 13
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (𝑆‘suc 𝑁) = ((𝑆‘𝑁) ∪ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁)))) |
30 | 29 | rexeqdv 3318 |
. . . . . . . . . . . 12
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ↔ ∃𝑢 ∈ ((𝑆‘𝑁) ∪ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁)))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)))) |
31 | | rexun 4081 |
. . . . . . . . . . . 12
⊢
(∃𝑢 ∈
((𝑆‘𝑁) ∪ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁)))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ↔ (∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)))) |
32 | 30, 31 | bitrdi 290 |
. . . . . . . . . . 11
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ↔ (∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))))) |
33 | 17, 32 | syl5bb 286 |
. . . . . . . . . 10
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)})) ↔ (∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))))) |
34 | | r19.43 3256 |
. . . . . . . . . . . . 13
⊢
(∃𝑢 ∈
(𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ↔ (∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))) |
35 | 22 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → 𝑁 ⊆ suc 𝑁) |
36 | 21, 35 | jca 515 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω)) ∧ 𝑁 ⊆ suc 𝑁)) |
37 | 36, 25 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (𝑆‘𝑁) ⊆ (𝑆‘suc 𝑁)) |
38 | 37 | adantr 484 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (𝑆‘𝑁) ⊆ (𝑆‘suc 𝑁)) |
39 | 38, 27 | sylib 221 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → ((𝑆‘𝑁) ∪ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))) = (𝑆‘suc 𝑁)) |
40 | 39 | eqcomd 2745 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (𝑆‘suc 𝑁) = ((𝑆‘𝑁) ∪ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁)))) |
41 | 40 | rexeqdv 3318 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ↔ ∃𝑣 ∈ ((𝑆‘𝑁) ∪ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁)))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))) |
42 | | rexun 4081 |
. . . . . . . . . . . . . . . . 17
⊢
(∃𝑣 ∈
((𝑆‘𝑁) ∪ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁)))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ↔ (∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))) |
43 | 41, 42 | bitrdi 290 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ↔ (∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))) |
44 | 43 | rexbidv 3208 |
. . . . . . . . . . . . . . 15
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ↔ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))) |
45 | 44 | orbi1d 916 |
. . . . . . . . . . . . . 14
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → ((∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ↔ (∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)))) |
46 | | r19.43 3256 |
. . . . . . . . . . . . . . . 16
⊢
(∃𝑢 ∈
(𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)) ↔ (∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))) |
47 | 46 | orbi1i 913 |
. . . . . . . . . . . . . . 15
⊢
((∃𝑢 ∈
(𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ↔ ((∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))) |
48 | | or32 925 |
. . . . . . . . . . . . . . . 16
⊢
(((∃𝑢 ∈
(𝑆‘𝑁)∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ↔ ((∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))) |
49 | | r19.43 3256 |
. . . . . . . . . . . . . . . . . 18
⊢
(∃𝑢 ∈
(𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ↔ (∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))) |
50 | 49 | bicomi 227 |
. . . . . . . . . . . . . . . . 17
⊢
((∃𝑢 ∈
(𝑆‘𝑁)∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ↔ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))) |
51 | 50 | orbi1i 913 |
. . . . . . . . . . . . . . . 16
⊢
(((∃𝑢 ∈
(𝑆‘𝑁)∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)) ↔ (∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))) |
52 | 48, 51 | bitri 278 |
. . . . . . . . . . . . . . 15
⊢
(((∃𝑢 ∈
(𝑆‘𝑁)∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ↔ (∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))) |
53 | 47, 52 | bitri 278 |
. . . . . . . . . . . . . 14
⊢
((∃𝑢 ∈
(𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ↔ (∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))) |
54 | 45, 53 | bitrdi 290 |
. . . . . . . . . . . . 13
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → ((∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ↔ (∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))) |
55 | 34, 54 | syl5bb 286 |
. . . . . . . . . . . 12
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ↔ (∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))) |
56 | | animorr 978 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) ∧ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))) → (⟨𝑥, 𝑦⟩ ∈ (𝑆‘𝑁) ∨ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)))) |
57 | 2 | satfvsuc 32897 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (𝑆‘suc 𝑁) = ((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))})) |
58 | 57 | eleq2d 2819 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (𝑠 ∈ (𝑆‘suc 𝑁) ↔ 𝑠 ∈ ((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))}))) |
59 | 58 | adantr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (𝑠 ∈ (𝑆‘suc 𝑁) ↔ 𝑠 ∈ ((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))}))) |
60 | | eleq1 2821 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 = ⟨𝑥, 𝑦⟩ → (𝑠 ∈ ((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))}) ↔ ⟨𝑥, 𝑦⟩ ∈ ((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))}))) |
61 | 60 | adantl 485 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (𝑠 ∈ ((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))}) ↔ ⟨𝑥, 𝑦⟩ ∈ ((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))}))) |
62 | | elun 4040 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(⟨𝑥, 𝑦⟩ ∈ ((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))}) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝑆‘𝑁) ∨ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))})) |
63 | | opabidw 5381 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))} ↔ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))) |
64 | 63 | orbi2i 912 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((⟨𝑥, 𝑦⟩ ∈ (𝑆‘𝑁) ∨ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))}) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝑆‘𝑁) ∨ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)})))) |
65 | 62, 64 | bitri 278 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(⟨𝑥, 𝑦⟩ ∈ ((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))}) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝑆‘𝑁) ∨ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)})))) |
66 | 61, 65 | bitrdi 290 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (𝑠 ∈ ((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))}) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝑆‘𝑁) ∨ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))))) |
67 | 59, 66 | bitrd 282 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (𝑠 ∈ (𝑆‘suc 𝑁) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝑆‘𝑁) ∨ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))))) |
68 | 7 | eqcomi 2748 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑀 ↑_{m} ω)
∖ ((2^{nd} ‘𝑢) ∩ (2^{nd} ‘𝑣))) = 𝐴 |
69 | 68 | eqeq2i 2752 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣))) ↔ 𝑦 = 𝐴) |
70 | 69 | anbi2i 626 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ↔ (𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)) |
71 | 70 | rexbii 3162 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∃𝑣 ∈
(𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ↔ ∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)) |
72 | 11 | eqcomi 2748 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)} = 𝐵 |
73 | 72 | eqeq2i 2752 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)} ↔ 𝑦 = 𝐵) |
74 | 73 | anbi2i 626 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑥 =
∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}) ↔ (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) |
75 | 74 | rexbii 3162 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∃𝑖 ∈
ω (𝑥 =
∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}) ↔ ∃𝑖 ∈ ω (𝑥 =
∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) |
76 | 71, 75 | orbi12i 914 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((∃𝑣 ∈
(𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)})) ↔ (∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))) |
77 | 76 | rexbii 3162 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∃𝑢 ∈
(𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)})) ↔ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))) |
78 | 77 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)})) ↔ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)))) |
79 | 78 | orbi2d 915 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → ((⟨𝑥, 𝑦⟩ ∈ (𝑆‘𝑁) ∨ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝑆‘𝑁) ∨ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))))) |
80 | 67, 79 | bitrd 282 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (𝑠 ∈ (𝑆‘suc 𝑁) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝑆‘𝑁) ∨ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))))) |
81 | 80 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) ∧ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))) → (𝑠 ∈ (𝑆‘suc 𝑁) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝑆‘𝑁) ∨ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))))) |
82 | 56, 81 | mpbird 260 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) ∧ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))) → 𝑠 ∈ (𝑆‘suc 𝑁)) |
83 | 82 | orcd 872 |
. . . . . . . . . . . . . 14
⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) ∧ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))) → (𝑠 ∈ (𝑆‘suc 𝑁) ∨ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))))) |
84 | 83 | ex 416 |
. . . . . . . . . . . . 13
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) → (𝑠 ∈ (𝑆‘suc 𝑁) ∨ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))))) |
85 | | simplr 769 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) ∧ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)) → 𝑠 = ⟨𝑥, 𝑦⟩) |
86 | | animorr 978 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) ∧ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)) → (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))) |
87 | 85, 86 | jca 515 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) ∧ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)) → (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))) |
88 | 87 | olcd 873 |
. . . . . . . . . . . . . 14
⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) ∧ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)) → (𝑠 ∈ (𝑆‘suc 𝑁) ∨ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))))) |
89 | 88 | ex 416 |
. . . . . . . . . . . . 13
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) → (𝑠 ∈ (𝑆‘suc 𝑁) ∨ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))))) |
90 | 84, 89 | jaod 858 |
. . . . . . . . . . . 12
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → ((∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)) → (𝑠 ∈ (𝑆‘suc 𝑁) ∨ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))))) |
91 | 55, 90 | sylbid 243 |
. . . . . . . . . . 11
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) → (𝑠 ∈ (𝑆‘suc 𝑁) ∨ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))))) |
92 | | simplr 769 |
. . . . . . . . . . . . . 14
⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) ∧ ∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))) → 𝑠 = ⟨𝑥, 𝑦⟩) |
93 | | orc 866 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑢 ∈
((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) → (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))) |
94 | 93 | adantl 485 |
. . . . . . . . . . . . . 14
⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) ∧ ∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))) → (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))) |
95 | 92, 94 | jca 515 |
. . . . . . . . . . . . 13
⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) ∧ ∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))) → (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))) |
96 | 95 | olcd 873 |
. . . . . . . . . . . 12
⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) ∧ ∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))) → (𝑠 ∈ (𝑆‘suc 𝑁) ∨ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))))) |
97 | 96 | ex 416 |
. . . . . . . . . . 11
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) → (𝑠 ∈ (𝑆‘suc 𝑁) ∨ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))))) |
98 | 91, 97 | jaod 858 |
. . . . . . . . . 10
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → ((∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))) → (𝑠 ∈ (𝑆‘suc 𝑁) ∨ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))))) |
99 | 33, 98 | sylbid 243 |
. . . . . . . . 9
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ 𝑠 = ⟨𝑥, 𝑦⟩) → (∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)})) → (𝑠 ∈ (𝑆‘suc 𝑁) ∨ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))))) |
100 | 99 | expimpd 457 |
. . . . . . . 8
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((𝑠 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))) → (𝑠 ∈ (𝑆‘suc 𝑁) ∨ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))))) |
101 | 100 | 2eximdv 1926 |
. . . . . . 7
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))) → ∃𝑥∃𝑦(𝑠 ∈ (𝑆‘suc 𝑁) ∨ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))))) |
102 | | 19.45v 2005 |
. . . . . . . . 9
⊢
(∃𝑦(𝑠 ∈ (𝑆‘suc 𝑁) ∨ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))) ↔ (𝑠 ∈ (𝑆‘suc 𝑁) ∨ ∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))))) |
103 | 102 | exbii 1854 |
. . . . . . . 8
⊢
(∃𝑥∃𝑦(𝑠 ∈ (𝑆‘suc 𝑁) ∨ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))) ↔ ∃𝑥(𝑠 ∈ (𝑆‘suc 𝑁) ∨ ∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))))) |
104 | | 19.45v 2005 |
. . . . . . . 8
⊢
(∃𝑥(𝑠 ∈ (𝑆‘suc 𝑁) ∨ ∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))) ↔ (𝑠 ∈ (𝑆‘suc 𝑁) ∨ ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))))) |
105 | 103, 104 | bitri 278 |
. . . . . . 7
⊢
(∃𝑥∃𝑦(𝑠 ∈ (𝑆‘suc 𝑁) ∨ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))) ↔ (𝑠 ∈ (𝑆‘suc 𝑁) ∨ ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))))) |
106 | 101, 105 | syl6ib 254 |
. . . . . 6
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))) → (𝑠 ∈ (𝑆‘suc 𝑁) ∨ ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))))) |
107 | 6, 106 | jaod 858 |
. . . . 5
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((𝑠 ∈ (𝑆‘suc 𝑁) ∨ ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)})))) → (𝑠 ∈ (𝑆‘suc 𝑁) ∨ ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))))) |
108 | | difss 4023 |
. . . . . . . . . . . 12
⊢ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁)) ⊆ (𝑆‘suc 𝑁) |
109 | | ssrexv 3945 |
. . . . . . . . . . . 12
⊢ (((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁)) ⊆ (𝑆‘suc 𝑁) → (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) → ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)))) |
110 | 108, 109 | ax-mp 5 |
. . . . . . . . . . 11
⊢
(∃𝑢 ∈
((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) → ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵))) |
111 | 110 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) → ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)))) |
112 | 111, 16 | syl6ib 254 |
. . . . . . . . 9
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) → ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)})))) |
113 | | ssrexv 3945 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑆‘𝑁) ⊆ (𝑆‘suc 𝑁) → (∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) → ∃𝑢 ∈ (𝑆‘suc 𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))) |
114 | 37, 113 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) → ∃𝑢 ∈ (𝑆‘suc 𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))) |
115 | 9 | 2rexbii 3163 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑢 ∈
(𝑆‘suc 𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ↔ ∃𝑢 ∈ (𝑆‘suc 𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣))))) |
116 | 114, 115 | syl6ib 254 |
. . . . . . . . . . . . . 14
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) → ∃𝑢 ∈ (𝑆‘suc 𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))))) |
117 | 116 | imp 410 |
. . . . . . . . . . . . 13
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)) → ∃𝑢 ∈ (𝑆‘suc 𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣))))) |
118 | | ssrexv 3945 |
. . . . . . . . . . . . . . 15
⊢ (((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁)) ⊆ (𝑆‘suc 𝑁) → (∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) → ∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))))) |
119 | 108, 118 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢
(∃𝑣 ∈
((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) → ∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣))))) |
120 | 119 | reximi 3158 |
. . . . . . . . . . . . 13
⊢
(∃𝑢 ∈
(𝑆‘suc 𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) → ∃𝑢 ∈ (𝑆‘suc 𝑁)∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣))))) |
121 | 117, 120 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)) → ∃𝑢 ∈ (𝑆‘suc 𝑁)∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣))))) |
122 | 121 | orcd 872 |
. . . . . . . . . . 11
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) ∧ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)) → (∃𝑢 ∈ (𝑆‘suc 𝑁)∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑢 ∈ (𝑆‘suc 𝑁)∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))) |
123 | 122 | ex 416 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) → (∃𝑢 ∈ (𝑆‘suc 𝑁)∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑢 ∈ (𝑆‘suc 𝑁)∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)})))) |
124 | | r19.43 3256 |
. . . . . . . . . 10
⊢
(∃𝑢 ∈
(𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)})) ↔ (∃𝑢 ∈ (𝑆‘suc 𝑁)∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑢 ∈ (𝑆‘suc 𝑁)∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))) |
125 | 123, 124 | syl6ibr 255 |
. . . . . . . . 9
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) → ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)})))) |
126 | 112, 125 | jaod 858 |
. . . . . . . 8
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)) → ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)})))) |
127 | 126 | anim2d 615 |
. . . . . . 7
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))) → (𝑠 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))))) |
128 | 127 | 2eximdv 1926 |
. . . . . 6
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))) → ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))))) |
129 | 128 | orim2d 966 |
. . . . 5
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((𝑠 ∈ (𝑆‘suc 𝑁) ∨ ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))) → (𝑠 ∈ (𝑆‘suc 𝑁) ∨ ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)})))))) |
130 | 107, 129 | impbid 215 |
. . . 4
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((𝑠 ∈ (𝑆‘suc 𝑁) ∨ ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)})))) ↔ (𝑠 ∈ (𝑆‘suc 𝑁) ∨ ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))))) |
131 | | elun 4040 |
. . . . 5
⊢ (𝑠 ∈ ((𝑆‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))}) ↔ (𝑠 ∈ (𝑆‘suc 𝑁) ∨ 𝑠 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))})) |
132 | | elopab 5383 |
. . . . . 6
⊢ (𝑠 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))} ↔ ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)})))) |
133 | 132 | orbi2i 912 |
. . . . 5
⊢ ((𝑠 ∈ (𝑆‘suc 𝑁) ∨ 𝑠 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))}) ↔ (𝑠 ∈ (𝑆‘suc 𝑁) ∨ ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))))) |
134 | 131, 133 | bitri 278 |
. . . 4
⊢ (𝑠 ∈ ((𝑆‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))}) ↔ (𝑠 ∈ (𝑆‘suc 𝑁) ∨ ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))))) |
135 | | elun 4040 |
. . . . 5
⊢ (𝑠 ∈ ((𝑆‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))}) ↔ (𝑠 ∈ (𝑆‘suc 𝑁) ∨ 𝑠 ∈ {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))})) |
136 | | elopab 5383 |
. . . . . 6
⊢ (𝑠 ∈ {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))} ↔ ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴)))) |
137 | 136 | orbi2i 912 |
. . . . 5
⊢ ((𝑠 ∈ (𝑆‘suc 𝑁) ∨ 𝑠 ∈ {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))}) ↔ (𝑠 ∈ (𝑆‘suc 𝑁) ∨ ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))))) |
138 | 135, 137 | bitri 278 |
. . . 4
⊢ (𝑠 ∈ ((𝑆‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))}) ↔ (𝑠 ∈ (𝑆‘suc 𝑁) ∨ ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))))) |
139 | 130, 134,
138 | 3bitr4g 317 |
. . 3
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (𝑠 ∈ ((𝑆‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))}) ↔ 𝑠 ∈ ((𝑆‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))}))) |
140 | 139 | eqrdv 2737 |
. 2
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((𝑆‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘suc 𝑁)(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_{m} ω) ∖
((2^{nd} ‘𝑢)
∩ (2^{nd} ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_{m} ω) ∣
∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^{nd} ‘𝑢)}))}) = ((𝑆‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))})) |
141 | 4, 140 | eqtrd 2774 |
1
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (𝑆‘suc suc 𝑁) = ((𝑆‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_{𝑔}𝑖(1^{st} ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1^{st} ‘𝑢)⊼_{𝑔}(1^{st}
‘𝑣)) ∧ 𝑦 = 𝐴))})) |