Proof of Theorem noetasuplem3
| Step | Hyp | Ref
| Expression |
| 1 | | simpl1 1192 |
. . . 4
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → 𝐴 ⊆ No
) |
| 2 | | simpl2 1193 |
. . . 4
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → 𝐴 ∈ V) |
| 3 | | simpr 484 |
. . . 4
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴) |
| 4 | | noetasuplem.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 27759 |
. . . 4
⊢ ((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝑋 ∈ 𝐴) → (𝑋 ↾ dom 𝑆) <s 𝑆) |
| 6 | 1, 2, 3, 5 | syl3anc 1373 |
. . 3
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → (𝑋 ↾ dom 𝑆) <s 𝑆) |
| 7 | | noetasuplem.2 |
. . . . . 6
⊢ 𝑍 = (𝑆 ∪ ((suc ∪
( bday “ 𝐵) ∖ dom 𝑆) × {1o})) |
| 8 | 7 | reseq1i 5993 |
. . . . 5
⊢ (𝑍 ↾ dom 𝑆) = ((𝑆 ∪ ((suc ∪
( bday “ 𝐵) ∖ dom 𝑆) × {1o})) ↾ dom
𝑆) |
| 9 | | resundir 6012 |
. . . . . 6
⊢ ((𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) ↾ dom
𝑆) = ((𝑆 ↾ dom 𝑆) ∪ (((suc ∪
( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆)) |
| 10 | | df-res 5697 |
. . . . . . . 8
⊢ (((suc
∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ∩ (dom 𝑆 × V)) |
| 11 | | disjdifr 4473 |
. . . . . . . . 9
⊢ ((suc
∪ ( bday “ 𝐵) ∖ dom 𝑆) ∩ dom 𝑆) = ∅ |
| 12 | | xpdisj1 6181 |
. . . . . . . . 9
⊢ (((suc
∪ ( bday “ 𝐵) ∖ dom 𝑆) ∩ dom 𝑆) = ∅ → (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ∩ (dom 𝑆 × V)) =
∅) |
| 13 | 11, 12 | ax-mp 5 |
. . . . . . . 8
⊢ (((suc
∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ∩ (dom 𝑆 × V)) =
∅ |
| 14 | 10, 13 | eqtri 2765 |
. . . . . . 7
⊢ (((suc
∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = ∅ |
| 15 | 14 | uneq2i 4165 |
. . . . . 6
⊢ ((𝑆 ↾ dom 𝑆) ∪ (((suc ∪
( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆)) = ((𝑆 ↾ dom 𝑆) ∪ ∅) |
| 16 | | un0 4394 |
. . . . . 6
⊢ ((𝑆 ↾ dom 𝑆) ∪ ∅) = (𝑆 ↾ dom 𝑆) |
| 17 | 9, 15, 16 | 3eqtri 2769 |
. . . . 5
⊢ ((𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) ↾ dom
𝑆) = (𝑆 ↾ dom 𝑆) |
| 18 | 8, 17 | eqtri 2765 |
. . . 4
⊢ (𝑍 ↾ dom 𝑆) = (𝑆 ↾ dom 𝑆) |
| 19 | 4 | nosupno 27748 |
. . . . . 6
⊢ ((𝐴 ⊆
No ∧ 𝐴 ∈
V) → 𝑆 ∈ No ) |
| 20 | 1, 2, 19 | syl2anc 584 |
. . . . 5
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → 𝑆 ∈ No
) |
| 21 | | nofun 27694 |
. . . . 5
⊢ (𝑆 ∈
No → Fun 𝑆) |
| 22 | | funrel 6583 |
. . . . 5
⊢ (Fun
𝑆 → Rel 𝑆) |
| 23 | | resdm 6044 |
. . . . 5
⊢ (Rel
𝑆 → (𝑆 ↾ dom 𝑆) = 𝑆) |
| 24 | 20, 21, 22, 23 | 4syl 19 |
. . . 4
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → (𝑆 ↾ dom 𝑆) = 𝑆) |
| 25 | 18, 24 | eqtrid 2789 |
. . 3
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → (𝑍 ↾ dom 𝑆) = 𝑆) |
| 26 | 6, 25 | breqtrrd 5171 |
. 2
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → (𝑋 ↾ dom 𝑆) <s (𝑍 ↾ dom 𝑆)) |
| 27 | | simp1 1137 |
. . . 4
⊢ ((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) →
𝐴 ⊆ No ) |
| 28 | 27 | sselda 3983 |
. . 3
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → 𝑋 ∈ No
) |
| 29 | 4, 7 | noetasuplem1 27778 |
. . . 4
⊢ ((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) →
𝑍 ∈ No ) |
| 30 | 29 | adantr 480 |
. . 3
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → 𝑍 ∈ No
) |
| 31 | | nodmon 27695 |
. . . 4
⊢ (𝑆 ∈
No → dom 𝑆
∈ On) |
| 32 | 20, 31 | syl 17 |
. . 3
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → dom 𝑆 ∈ On) |
| 33 | | sltres 27707 |
. . 3
⊢ ((𝑋 ∈
No ∧ 𝑍 ∈
No ∧ dom 𝑆 ∈ On) → ((𝑋 ↾ dom 𝑆) <s (𝑍 ↾ dom 𝑆) → 𝑋 <s 𝑍)) |
| 34 | 28, 30, 32, 33 | syl3anc 1373 |
. 2
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → ((𝑋 ↾ dom 𝑆) <s (𝑍 ↾ dom 𝑆) → 𝑋 <s 𝑍)) |
| 35 | 26, 34 | mpd 15 |
1
⊢ (((𝐴 ⊆
No ∧ 𝐴 ∈ V
∧ 𝐵 ∈ V) ∧
𝑋 ∈ 𝐴) → 𝑋 <s 𝑍) |