Theorem funsneqopb 7024

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 4804	. . . . . 6 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐵⟩)
3	2	sneqd 4573	. . . . 5 ⊢ (𝐴 = 𝐵 → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐵⟩})
4		funsndifnop.b	. . . . . 6 ⊢ 𝐵 ∈ V
5	4	snopeqopsnid 5423	. . . . 5 ⊢ {⟨𝐵, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩
6	3, 5	eqtrdi 2794	. . . 4 ⊢ (𝐴 = 𝐵 → {⟨𝐴, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩)
7	1, 6	eqtrid 2790	. . 3 ⊢ (𝐴 = 𝐵 → 𝐺 = ⟨{𝐵}, {𝐵}⟩)
8		snex 5354	. . . 4 ⊢ {𝐵} ∈ V
9	8, 8	opelvv 5628	. . 3 ⊢ ⟨{𝐵}, {𝐵}⟩ ∈ (V × V)
10	7, 9	eqeltrdi 2847	. 2 ⊢ (𝐴 = 𝐵 → 𝐺 ∈ (V × V))
11		funsndifnop.a	. . . 4 ⊢ 𝐴 ∈ V
12	11, 4, 1	funsndifnop 7023	. . 3 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐺 ∈ (V × V))
13	12	necon4ai 2975	. 2 ⊢ (𝐺 ∈ (V × V) → 𝐴 = 𝐵)
14	10, 13	impbii 208	1 ⊢ (𝐴 = 𝐵 ↔ 𝐺 ∈ (V × V))

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2106  Vcvv 3432  {csn 4561  ⟨cop 4567   × cxp 5587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441

This theorem is referenced by: (None)