Theorem funsneqopb 7086

Description: A singleton of an ordered pair is an ordered pair iff the components are equal. (Contributed by AV, 24-Sep-2020.) (Avoid depending on this detail.)

Hypotheses

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

Assertion

Ref	Expression
funsneqopb	⊢ (𝐴 = 𝐵 ↔ 𝐺 ∈ (V × V))

Proof of Theorem funsneqopb

Step	Hyp	Ref	Expression
1		funsndifnop.g	. . . 4 ⊢ 𝐺 = {⟨𝐴, 𝐵⟩}
2		opeq1 4824	. . . . . 6 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐵⟩)
3	2	sneqd 4589	. . . . 5 ⊢ (𝐴 = 𝐵 → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐵⟩})
4		funsndifnop.b	. . . . . 6 ⊢ 𝐵 ∈ V
5	4	snopeqopsnid 5452	. . . . 5 ⊢ {⟨𝐵, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩
6	3, 5	eqtrdi 2780	. . . 4 ⊢ (𝐴 = 𝐵 → {⟨𝐴, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩)
7	1, 6	eqtrid 2776	. . 3 ⊢ (𝐴 = 𝐵 → 𝐺 = ⟨{𝐵}, {𝐵}⟩)
8		snex 5375	. . . 4 ⊢ {𝐵} ∈ V
9	8, 8	opelvv 5659	. . 3 ⊢ ⟨{𝐵}, {𝐵}⟩ ∈ (V × V)
10	7, 9	eqeltrdi 2836	. 2 ⊢ (𝐴 = 𝐵 → 𝐺 ∈ (V × V))
11		funsndifnop.a	. . . 4 ⊢ 𝐴 ∈ V
12	11, 4, 1	funsndifnop 7085	. . 3 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐺 ∈ (V × V))
13	12	necon4ai 2956	. 2 ⊢ (𝐺 ∈ (V × V) → 𝐴 = 𝐵)
14	10, 13	impbii 209	1 ⊢ (𝐴 = 𝐵 ↔ 𝐺 ∈ (V × V))

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  Vcvv 3436  {csn 4577  ⟨cop 4583   × cxp 5617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490

This theorem is referenced by: (None)