Theorem funsneqopb 7106

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 4816	. . . . . 6 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐵⟩)
3	2	sneqd 4579	. . . . 5 ⊢ (𝐴 = 𝐵 → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐵⟩})
4		funsndifnop.b	. . . . . 6 ⊢ 𝐵 ∈ V
5	4	snopeqopsnid 5463	. . . . 5 ⊢ {⟨𝐵, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩
6	3, 5	eqtrdi 2787	. . . 4 ⊢ (𝐴 = 𝐵 → {⟨𝐴, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩)
7	1, 6	eqtrid 2783	. . 3 ⊢ (𝐴 = 𝐵 → 𝐺 = ⟨{𝐵}, {𝐵}⟩)
8		snex 5381	. . . 4 ⊢ {𝐵} ∈ V
9	8, 8	opelvv 5671	. . 3 ⊢ ⟨{𝐵}, {𝐵}⟩ ∈ (V × V)
10	7, 9	eqeltrdi 2844	. 2 ⊢ (𝐴 = 𝐵 → 𝐺 ∈ (V × V))
11		funsndifnop.a	. . . 4 ⊢ 𝐴 ∈ V
12	11, 4, 1	funsndifnop 7105	. . 3 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐺 ∈ (V × V))
13	12	necon4ai 2963	. 2 ⊢ (𝐺 ∈ (V × V) → 𝐴 = 𝐵)
14	10, 13	impbii 209	1 ⊢ (𝐴 = 𝐵 ↔ 𝐺 ∈ (V × V))

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  Vcvv 3429  {csn 4567  ⟨cop 4573   × cxp 5629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506

This theorem is referenced by: (None)