Theorem en2snOLD 9073

Description: Obsolete version of en2sn 9072 as of 25-Sep-2024. (Contributed by NM, 9-Nov-2003.) Avoid ax-pow 5369. (Revised by BTernaryTau, 31-Jul-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
en2snOLD	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵})

Proof of Theorem en2snOLD

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 5437	. . 3 ⊢ {⟨𝐴, 𝐵⟩} ∈ V
2		f1osng 6885	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})
3		f1oeq1 6832	. . . 4 ⊢ (𝑓 = {⟨𝐴, 𝐵⟩} → (𝑓:{𝐴}–1-1-onto→{𝐵} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
4	3	spcegv 3586	. . 3 ⊢ ({⟨𝐴, 𝐵⟩} ∈ V → ({⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵} → ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵}))
5	1, 2, 4	mpsyl 68	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵})
6		bren 8980	. 2 ⊢ ({𝐴} ≈ {𝐵} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵})
7	5, 6	sylibr 233	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵})

Syntax hints:   → wi 4   ∧ wa 394  ∃wex 1773   ∈ wcel 2098  Vcvv 3473  {csn 4632  ⟨cop 4638   class class class wbr 5152  –1-1-onto→wf1o 6552   ≈ cen 8967

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-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2529  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-en 8971

This theorem is referenced by: (None)