Theorem wevonprcf1o 35420

Description: If 𝑅 is a set-like well-ordering of the universe and 𝐴 is a proper class, then 𝐹 is a bijection from the ordinals to 𝐴. This is the ZFC version of (4 → 5) in https://tinyurl.com/hamkins-gblac. (Contributed by BTernaryTau, 9-Jun-2026.)

Hypothesis

Ref	Expression
wevonprcf1o.1	⊢ 𝐹 = OrdIso(𝑅, 𝐴)

Assertion

Ref	Expression
wevonprcf1o	⊢ ((𝑅 We V ∧ 𝑅 Se V ∧ ¬ 𝐴 ∈ V) → 𝐹:On–1-1-onto→𝐴)

Proof of Theorem wevonprcf1o

Step	Hyp	Ref	Expression
1		ssv 3960	. . . 4 ⊢ 𝐴 ⊆ V
2		wess 5631	. . . 4 ⊢ (𝐴 ⊆ V → (𝑅 We V → 𝑅 We 𝐴))
3	1, 2	ax-mp 5	. . 3 ⊢ (𝑅 We V → 𝑅 We 𝐴)
4		sess2 5611	. . . 4 ⊢ (𝐴 ⊆ V → (𝑅 Se V → 𝑅 Se 𝐴))
5	1, 4	ax-mp 5	. . 3 ⊢ (𝑅 Se V → 𝑅 Se 𝐴)
6		id 22	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ V)
7	3, 5, 6	3anim123i 1163	. 2 ⊢ ((𝑅 We V ∧ 𝑅 Se V ∧ ¬ 𝐴 ∈ V) → (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ¬ 𝐴 ∈ V))
8		wevonprcf1o.1	. . 3 ⊢ 𝐹 = OrdIso(𝑅, 𝐴)
9	8	ordtypeon 35350	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ¬ 𝐴 ∈ V) → 𝐹 Isom E , 𝑅 (On, 𝐴))
10		isof1o 7303	. 2 ⊢ (𝐹 Isom E , 𝑅 (On, 𝐴) → 𝐹:On–1-1-onto→𝐴)
11	7, 9, 10	3syl 18	1 ⊢ ((𝑅 We V ∧ 𝑅 Se V ∧ ¬ 𝐴 ∈ V) → 𝐹:On–1-1-onto→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3904   E cep 5544   Se wse 5596   We wwe 5597  Oncon0 6342  –1-1-onto→wf1o 6516   Isom wiso 6518  OrdIsocoi 9454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-oi 9455

This theorem is referenced by: (None)