Theorem funsneqopOLD 6670

Description: Obsolete as of 15-Jul-2022. (Contributed by AV, 24-Sep-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
funsndifnop.a	⊢ 𝐴 ∈ V
funsndifnop.b	⊢ 𝐵 ∈ V
funsndifnop.g	⊢ 𝐺 = {⟨𝐴, 𝐵⟩}

Assertion

Ref	Expression
funsneqopOLD	⊢ (𝐴 = 𝐵 → 𝐺 ∈ (V × V))

Proof of Theorem funsneqopOLD

Step	Hyp	Ref	Expression
1		funsndifnop.a	. . 3 ⊢ 𝐴 ∈ V
2		funsndifnop.b	. . 3 ⊢ 𝐵 ∈ V
3		funsndifnop.g	. . 3 ⊢ 𝐺 = {⟨𝐴, 𝐵⟩}
4	1, 2, 3	funsneqopsnOLD 6669	. 2 ⊢ (𝐴 = 𝐵 → 𝐺 = ⟨{𝐴}, {𝐴}⟩)
5		snex 5130	. . 3 ⊢ {𝐴} ∈ V
6	5, 5	opelvv 5382	. 2 ⊢ ⟨{𝐴}, {𝐴}⟩ ∈ (V × V)
7	4, 6	syl6eqel 2915	1 ⊢ (𝐴 = 𝐵 → 𝐺 ∈ (V × V))

Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166  Vcvv 3415  {csn 4398  ⟨cop 4404   × cxp 5341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-opab 4937  df-xp 5349

This theorem is referenced by: (None)