Theorem ordiso 9236

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

Step	Hyp	Ref	Expression
1		resiexg 7748	. . . . 5 ⊢ (𝐴 ∈ On → ( I ↾ 𝐴) ∈ V)
2		isoid 7193	. . . . 5 ⊢ ( I ↾ 𝐴) Isom E , E (𝐴, 𝐴)
3		isoeq1 7181	. . . . . 6 ⊢ (𝑓 = ( I ↾ 𝐴) → (𝑓 Isom E , E (𝐴, 𝐴) ↔ ( I ↾ 𝐴) Isom E , E (𝐴, 𝐴)))
4	3	spcegv 3534	. . . . 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 480	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∃𝑓 𝑓 Isom E , E (𝐴, 𝐴))
7		isoeq5 7185	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑓 Isom E , E (𝐴, 𝐴) ↔ 𝑓 Isom E , E (𝐴, 𝐵)))
8	7	exbidv 1927	. . 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 6273	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
11		eloni 6273	. . . 4 ⊢ (𝐵 ∈ On → Ord 𝐵)
12		ordiso2 9235	. . . . . 6 ⊢ ((𝑓 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵)
13	12	3coml 1125	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ 𝑓 Isom E , E (𝐴, 𝐵)) → 𝐴 = 𝐵)
14	13	3expia 1119	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝑓 Isom E , E (𝐴, 𝐵) → 𝐴 = 𝐵))
15	10, 11, 14	syl2an 595	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑓 Isom E , E (𝐴, 𝐵) → 𝐴 = 𝐵))
16	15	exlimdv 1939	. 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)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1541  ∃wex 1785   ∈ wcel 2109  Vcvv 3430   I cid 5487   E cep 5493   ↾ cres 5590  Ord word 6262  Oncon0 6263   Isom wiso 6431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-ord 6266  df-on 6267  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-isom 6439

This theorem is referenced by: (None)