Proof of Theorem noetalem2
Step | Hyp | Ref
| Expression |
1 | | simpl1 1184 |
. . . 4
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → 𝐴 ⊆ No
) |
2 | | simpl2 1185 |
. . . 4
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → 𝐴 ∈ V) |
3 | | simpr 485 |
. . . 4
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴) |
4 | | noetalem.1 |
. . . . 5
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) |
5 | 4 | nosupbnd1 32825 |
. . . 4
⊢ ((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝑋 ∈ 𝐴) → (𝑋 ↾ dom 𝑆) <s 𝑆) |
6 | 1, 2, 3, 5 | syl3anc 1364 |
. . 3
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → (𝑋 ↾ dom 𝑆) <s 𝑆) |
7 | | noetalem.2 |
. . . . . 6
⊢ 𝑍 = (𝑆 ∪ ((suc ∪
( bday “ 𝐵) ∖ dom 𝑆) × {1o})) |
8 | 7 | reseq1i 5737 |
. . . . 5
⊢ (𝑍 ↾ dom 𝑆) = ((𝑆 ∪ ((suc ∪
( bday “ 𝐵) ∖ dom 𝑆) × {1o})) ↾ dom
𝑆) |
9 | | resundir 5756 |
. . . . . 6
⊢ ((𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) ↾ dom
𝑆) = ((𝑆 ↾ dom 𝑆) ∪ (((suc ∪
( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆)) |
10 | | df-res 5462 |
. . . . . . . 8
⊢ (((suc
∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ∩ (dom 𝑆 × V)) |
11 | | incom 4105 |
. . . . . . . . . 10
⊢ ((suc
∪ ( bday “ 𝐵) ∖ dom 𝑆) ∩ dom 𝑆) = (dom 𝑆 ∩ (suc ∪
( bday “ 𝐵) ∖ dom 𝑆)) |
12 | | disjdif 4341 |
. . . . . . . . . 10
⊢ (dom
𝑆 ∩ (suc ∪ ( bday “ 𝐵) ∖ dom 𝑆)) = ∅ |
13 | 11, 12 | eqtri 2821 |
. . . . . . . . 9
⊢ ((suc
∪ ( bday “ 𝐵) ∖ dom 𝑆) ∩ dom 𝑆) = ∅ |
14 | | xpdisj1 5901 |
. . . . . . . . 9
⊢ (((suc
∪ ( bday “ 𝐵) ∖ dom 𝑆) ∩ dom 𝑆) = ∅ → (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ∩ (dom 𝑆 × V)) =
∅) |
15 | 13, 14 | ax-mp 5 |
. . . . . . . 8
⊢ (((suc
∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ∩ (dom 𝑆 × V)) =
∅ |
16 | 10, 15 | eqtri 2821 |
. . . . . . 7
⊢ (((suc
∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = ∅ |
17 | 16 | uneq2i 4063 |
. . . . . 6
⊢ ((𝑆 ↾ dom 𝑆) ∪ (((suc ∪
( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆)) = ((𝑆 ↾ dom 𝑆) ∪ ∅) |
18 | | un0 4270 |
. . . . . 6
⊢ ((𝑆 ↾ dom 𝑆) ∪ ∅) = (𝑆 ↾ dom 𝑆) |
19 | 9, 17, 18 | 3eqtri 2825 |
. . . . 5
⊢ ((𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) ↾ dom
𝑆) = (𝑆 ↾ dom 𝑆) |
20 | 8, 19 | eqtri 2821 |
. . . 4
⊢ (𝑍 ↾ dom 𝑆) = (𝑆 ↾ dom 𝑆) |
21 | 4 | nosupno 32814 |
. . . . . . 7
⊢ ((𝐴 ⊆
No ∧ 𝐴 ∈
V) → 𝑆 ∈ No ) |
22 | 1, 2, 21 | syl2anc 584 |
. . . . . 6
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → 𝑆 ∈ No
) |
23 | | nofun 32767 |
. . . . . 6
⊢ (𝑆 ∈
No → Fun 𝑆) |
24 | 22, 23 | syl 17 |
. . . . 5
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → Fun 𝑆) |
25 | | funrel 6249 |
. . . . 5
⊢ (Fun
𝑆 → Rel 𝑆) |
26 | | resdm 5785 |
. . . . 5
⊢ (Rel
𝑆 → (𝑆 ↾ dom 𝑆) = 𝑆) |
27 | 24, 25, 26 | 3syl 18 |
. . . 4
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → (𝑆 ↾ dom 𝑆) = 𝑆) |
28 | 20, 27 | syl5eq 2845 |
. . 3
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → (𝑍 ↾ dom 𝑆) = 𝑆) |
29 | 6, 28 | breqtrrd 4996 |
. 2
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → (𝑋 ↾ dom 𝑆) <s (𝑍 ↾ dom 𝑆)) |
30 | | simp1 1129 |
. . . 4
⊢ ((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) →
𝐴 ⊆ No ) |
31 | 30 | sselda 3895 |
. . 3
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → 𝑋 ∈ No
) |
32 | 4, 7 | noetalem1 32828 |
. . . 4
⊢ ((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) →
𝑍 ∈ No ) |
33 | 32 | adantr 481 |
. . 3
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → 𝑍 ∈ No
) |
34 | | nodmon 32768 |
. . . 4
⊢ (𝑆 ∈
No → dom 𝑆
∈ On) |
35 | 22, 34 | syl 17 |
. . 3
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → dom 𝑆 ∈ On) |
36 | | sltres 32780 |
. . 3
⊢ ((𝑋 ∈
No ∧ 𝑍 ∈
No ∧ dom 𝑆 ∈ On) → ((𝑋 ↾ dom 𝑆) <s (𝑍 ↾ dom 𝑆) → 𝑋 <s 𝑍)) |
37 | 31, 33, 35, 36 | syl3anc 1364 |
. 2
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → ((𝑋 ↾ dom 𝑆) <s (𝑍 ↾ dom 𝑆) → 𝑋 <s 𝑍)) |
38 | 29, 37 | mpd 15 |
1
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → 𝑋 <s 𝑍) |