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Theorem ordiso 8968
 Description: Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
ordiso ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = 𝐵 ↔ ∃𝑓 𝑓 Isom E , E (𝐴, 𝐵)))
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem ordiso
StepHypRef Expression
1 resiexg 7605 . . . . 5 (𝐴 ∈ On → ( I ↾ 𝐴) ∈ V)
2 isoid 7065 . . . . 5 ( I ↾ 𝐴) Isom E , E (𝐴, 𝐴)
3 isoeq1 7053 . . . . . 6 (𝑓 = ( I ↾ 𝐴) → (𝑓 Isom E , E (𝐴, 𝐴) ↔ ( I ↾ 𝐴) Isom E , E (𝐴, 𝐴)))
43spcegv 3548 . . . . 5 (( I ↾ 𝐴) ∈ V → (( I ↾ 𝐴) Isom E , E (𝐴, 𝐴) → ∃𝑓 𝑓 Isom E , E (𝐴, 𝐴)))
51, 2, 4mpisyl 21 . . . 4 (𝐴 ∈ On → ∃𝑓 𝑓 Isom E , E (𝐴, 𝐴))
65adantr 484 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∃𝑓 𝑓 Isom E , E (𝐴, 𝐴))
7 isoeq5 7057 . . . 4 (𝐴 = 𝐵 → (𝑓 Isom E , E (𝐴, 𝐴) ↔ 𝑓 Isom E , E (𝐴, 𝐵)))
87exbidv 1922 . . 3 (𝐴 = 𝐵 → (∃𝑓 𝑓 Isom E , E (𝐴, 𝐴) ↔ ∃𝑓 𝑓 Isom E , E (𝐴, 𝐵)))
96, 8syl5ibcom 248 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = 𝐵 → ∃𝑓 𝑓 Isom E , E (𝐴, 𝐵)))
10 eloni 6173 . . . 4 (𝐴 ∈ On → Ord 𝐴)
11 eloni 6173 . . . 4 (𝐵 ∈ On → Ord 𝐵)
12 ordiso2 8967 . . . . . 6 ((𝑓 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵)
13123coml 1124 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵𝑓 Isom E , E (𝐴, 𝐵)) → 𝐴 = 𝐵)
14133expia 1118 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑓 Isom E , E (𝐴, 𝐵) → 𝐴 = 𝐵))
1510, 11, 14syl2an 598 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑓 Isom E , E (𝐴, 𝐵) → 𝐴 = 𝐵))
1615exlimdv 1934 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓 𝑓 Isom E , E (𝐴, 𝐵) → 𝐴 = 𝐵))
179, 16impbid 215 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 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2112  Vcvv 3444   I cid 5427   E cep 5432   ↾ cres 5525  Ord word 6162  Oncon0 6163   Isom wiso 6329 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337 This theorem is referenced by: (None)
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