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Mirrors > Home > MPE Home > Th. List > ordiso | Structured version Visualization version GIF version |
Description: Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
ordiso | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = 𝐵 ↔ ∃𝑓 𝑓 Isom E , E (𝐴, 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resiexg 7920 | . . . . 5 ⊢ (𝐴 ∈ On → ( I ↾ 𝐴) ∈ V) | |
2 | isoid 7336 | . . . . 5 ⊢ ( I ↾ 𝐴) Isom E , E (𝐴, 𝐴) | |
3 | isoeq1 7324 | . . . . . 6 ⊢ (𝑓 = ( I ↾ 𝐴) → (𝑓 Isom E , E (𝐴, 𝐴) ↔ ( I ↾ 𝐴) Isom E , E (𝐴, 𝐴))) | |
4 | 3 | spcegv 3581 | . . . . 5 ⊢ (( I ↾ 𝐴) ∈ V → (( I ↾ 𝐴) Isom E , E (𝐴, 𝐴) → ∃𝑓 𝑓 Isom E , E (𝐴, 𝐴))) |
5 | 1, 2, 4 | mpisyl 21 | . . . 4 ⊢ (𝐴 ∈ On → ∃𝑓 𝑓 Isom E , E (𝐴, 𝐴)) |
6 | 5 | adantr 479 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∃𝑓 𝑓 Isom E , E (𝐴, 𝐴)) |
7 | isoeq5 7328 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑓 Isom E , E (𝐴, 𝐴) ↔ 𝑓 Isom E , E (𝐴, 𝐵))) | |
8 | 7 | exbidv 1916 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑓 𝑓 Isom E , E (𝐴, 𝐴) ↔ ∃𝑓 𝑓 Isom E , E (𝐴, 𝐵))) |
9 | 6, 8 | syl5ibcom 244 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = 𝐵 → ∃𝑓 𝑓 Isom E , E (𝐴, 𝐵))) |
10 | eloni 6381 | . . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
11 | eloni 6381 | . . . 4 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
12 | ordiso2 9540 | . . . . . 6 ⊢ ((𝑓 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵) | |
13 | 12 | 3coml 1124 | . . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ 𝑓 Isom E , E (𝐴, 𝐵)) → 𝐴 = 𝐵) |
14 | 13 | 3expia 1118 | . . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝑓 Isom E , E (𝐴, 𝐵) → 𝐴 = 𝐵)) |
15 | 10, 11, 14 | syl2an 594 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑓 Isom E , E (𝐴, 𝐵) → 𝐴 = 𝐵)) |
16 | 15 | exlimdv 1928 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓 𝑓 Isom E , E (𝐴, 𝐵) → 𝐴 = 𝐵)) |
17 | 9, 16 | impbid 211 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = 𝐵 ↔ ∃𝑓 𝑓 Isom E , E (𝐴, 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 Vcvv 3461 I cid 5575 E cep 5581 ↾ cres 5680 Ord word 6370 Oncon0 6371 Isom wiso 6550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6374 df-on 6375 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 |
This theorem is referenced by: (None) |
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