Step | Hyp | Ref
| Expression |
1 | | undifss 3495 |
. . . . . . . 8
⊢ (𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) ⊆ 𝑦) |
2 | 1 | biimpi 119 |
. . . . . . 7
⊢ (𝑥 ⊆ 𝑦 → (𝑥 ∪ (𝑦 ∖ 𝑥)) ⊆ 𝑦) |
3 | 2 | adantl 275 |
. . . . . 6
⊢
((EXMID ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∪ (𝑦 ∖ 𝑥)) ⊆ 𝑦) |
4 | | elun1 3294 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ 𝑥 → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥))) |
5 | 4 | adantl 275 |
. . . . . . . . . 10
⊢
(((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥))) |
6 | | simplr 525 |
. . . . . . . . . . . 12
⊢
(((EXMID ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑧 ∈ 𝑦) |
7 | | simpr 109 |
. . . . . . . . . . . 12
⊢
(((EXMID ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 ∈ 𝑥) → ¬ 𝑧 ∈ 𝑥) |
8 | 6, 7 | eldifd 3131 |
. . . . . . . . . . 11
⊢
(((EXMID ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑧 ∈ (𝑦 ∖ 𝑥)) |
9 | | elun2 3295 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ (𝑦 ∖ 𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥))) |
10 | 8, 9 | syl 14 |
. . . . . . . . . 10
⊢
(((EXMID ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥))) |
11 | | exmidexmid 4182 |
. . . . . . . . . . . 12
⊢
(EXMID → DECID 𝑧 ∈ 𝑥) |
12 | | exmiddc 831 |
. . . . . . . . . . . 12
⊢
(DECID 𝑧 ∈ 𝑥 → (𝑧 ∈ 𝑥 ∨ ¬ 𝑧 ∈ 𝑥)) |
13 | 11, 12 | syl 14 |
. . . . . . . . . . 11
⊢
(EXMID → (𝑧 ∈ 𝑥 ∨ ¬ 𝑧 ∈ 𝑥)) |
14 | 13 | adantr 274 |
. . . . . . . . . 10
⊢
((EXMID ∧ 𝑧 ∈ 𝑦) → (𝑧 ∈ 𝑥 ∨ ¬ 𝑧 ∈ 𝑥)) |
15 | 5, 10, 14 | mpjaodan 793 |
. . . . . . . . 9
⊢
((EXMID ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥))) |
16 | 15 | ex 114 |
. . . . . . . 8
⊢
(EXMID → (𝑧 ∈ 𝑦 → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥)))) |
17 | 16 | ssrdv 3153 |
. . . . . . 7
⊢
(EXMID → 𝑦 ⊆ (𝑥 ∪ (𝑦 ∖ 𝑥))) |
18 | 17 | adantr 274 |
. . . . . 6
⊢
((EXMID ∧ 𝑥 ⊆ 𝑦) → 𝑦 ⊆ (𝑥 ∪ (𝑦 ∖ 𝑥))) |
19 | 3, 18 | eqssd 3164 |
. . . . 5
⊢
((EXMID ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) |
20 | 19 | ex 114 |
. . . 4
⊢
(EXMID → (𝑥 ⊆ 𝑦 → (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦)) |
21 | | ssun1 3290 |
. . . . 5
⊢ 𝑥 ⊆ (𝑥 ∪ (𝑦 ∖ 𝑥)) |
22 | | sseq2 3171 |
. . . . 5
⊢ ((𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦 → (𝑥 ⊆ (𝑥 ∪ (𝑦 ∖ 𝑥)) ↔ 𝑥 ⊆ 𝑦)) |
23 | 21, 22 | mpbii 147 |
. . . 4
⊢ ((𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦 → 𝑥 ⊆ 𝑦) |
24 | 20, 23 | impbid1 141 |
. . 3
⊢
(EXMID → (𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦)) |
25 | 24 | alrimivv 1868 |
. 2
⊢
(EXMID → ∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦)) |
26 | | vex 2733 |
. . . . . 6
⊢ 𝑧 ∈ V |
27 | | p0ex 4174 |
. . . . . 6
⊢ {∅}
∈ V |
28 | | sseq12 3172 |
. . . . . . . 8
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → (𝑥 ⊆ 𝑦 ↔ 𝑧 ⊆ {∅})) |
29 | | simpl 108 |
. . . . . . . . . 10
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → 𝑥 = 𝑧) |
30 | | simpr 109 |
. . . . . . . . . . 11
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → 𝑦 = {∅}) |
31 | 30, 29 | difeq12d 3246 |
. . . . . . . . . 10
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → (𝑦 ∖ 𝑥) = ({∅} ∖ 𝑧)) |
32 | 29, 31 | uneq12d 3282 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → (𝑥 ∪ (𝑦 ∖ 𝑥)) = (𝑧 ∪ ({∅} ∖ 𝑧))) |
33 | 32, 30 | eqeq12d 2185 |
. . . . . . . 8
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → ((𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦 ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅})) |
34 | 28, 33 | bibi12d 234 |
. . . . . . 7
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → ((𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) ↔ (𝑧 ⊆ {∅} ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅}))) |
35 | 34 | spc2gv 2821 |
. . . . . 6
⊢ ((𝑧 ∈ V ∧ {∅} ∈
V) → (∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) → (𝑧 ⊆ {∅} ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅}))) |
36 | 26, 27, 35 | mp2an 424 |
. . . . 5
⊢
(∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) → (𝑧 ⊆ {∅} ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅})) |
37 | | 0ex 4116 |
. . . . . . . 8
⊢ ∅
∈ V |
38 | 37 | snid 3614 |
. . . . . . 7
⊢ ∅
∈ {∅} |
39 | | eleq2 2234 |
. . . . . . 7
⊢ ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → (∅
∈ (𝑧 ∪ ({∅}
∖ 𝑧)) ↔ ∅
∈ {∅})) |
40 | 38, 39 | mpbiri 167 |
. . . . . 6
⊢ ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → ∅
∈ (𝑧 ∪ ({∅}
∖ 𝑧))) |
41 | | eldifn 3250 |
. . . . . . . 8
⊢ (∅
∈ ({∅} ∖ 𝑧) → ¬ ∅ ∈ 𝑧) |
42 | 41 | orim2i 756 |
. . . . . . 7
⊢ ((∅
∈ 𝑧 ∨ ∅
∈ ({∅} ∖ 𝑧)) → (∅ ∈ 𝑧 ∨ ¬ ∅ ∈ 𝑧)) |
43 | | elun 3268 |
. . . . . . 7
⊢ (∅
∈ (𝑧 ∪ ({∅}
∖ 𝑧)) ↔ (∅
∈ 𝑧 ∨ ∅
∈ ({∅} ∖ 𝑧))) |
44 | | df-dc 830 |
. . . . . . 7
⊢
(DECID ∅ ∈ 𝑧 ↔ (∅ ∈ 𝑧 ∨ ¬ ∅ ∈ 𝑧)) |
45 | 42, 43, 44 | 3imtr4i 200 |
. . . . . 6
⊢ (∅
∈ (𝑧 ∪ ({∅}
∖ 𝑧)) →
DECID ∅ ∈ 𝑧) |
46 | 40, 45 | syl 14 |
. . . . 5
⊢ ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} →
DECID ∅ ∈ 𝑧) |
47 | 36, 46 | syl6bi 162 |
. . . 4
⊢
(∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) → (𝑧 ⊆ {∅} →
DECID ∅ ∈ 𝑧)) |
48 | 47 | alrimiv 1867 |
. . 3
⊢
(∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) → ∀𝑧(𝑧 ⊆ {∅} →
DECID ∅ ∈ 𝑧)) |
49 | | df-exmid 4181 |
. . 3
⊢
(EXMID ↔ ∀𝑧(𝑧 ⊆ {∅} →
DECID ∅ ∈ 𝑧)) |
50 | 48, 49 | sylibr 133 |
. 2
⊢
(∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) →
EXMID) |
51 | 25, 50 | impbii 125 |
1
⊢
(EXMID ↔ ∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦)) |