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Theorem funsneqopb 7172
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 = 𝐵𝐺 ∈ (V × V))

Proof of Theorem funsneqopb
StepHypRef Expression
1 funsndifnop.g . . . 4 𝐺 = {⟨𝐴, 𝐵⟩}
2 opeq1 4878 . . . . . 6 (𝐴 = 𝐵 → ⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐵⟩)
32sneqd 4643 . . . . 5 (𝐴 = 𝐵 → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐵⟩})
4 funsndifnop.b . . . . . 6 𝐵 ∈ V
54snopeqopsnid 5519 . . . . 5 {⟨𝐵, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩
63, 5eqtrdi 2791 . . . 4 (𝐴 = 𝐵 → {⟨𝐴, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩)
71, 6eqtrid 2787 . . 3 (𝐴 = 𝐵𝐺 = ⟨{𝐵}, {𝐵}⟩)
8 snex 5442 . . . 4 {𝐵} ∈ V
98, 8opelvv 5729 . . 3 ⟨{𝐵}, {𝐵}⟩ ∈ (V × V)
107, 9eqeltrdi 2847 . 2 (𝐴 = 𝐵𝐺 ∈ (V × V))
11 funsndifnop.a . . . 4 𝐴 ∈ V
1211, 4, 1funsndifnop 7171 . . 3 (𝐴𝐵 → ¬ 𝐺 ∈ (V × V))
1312necon4ai 2970 . 2 (𝐺 ∈ (V × V) → 𝐴 = 𝐵)
1410, 13impbii 209 1 (𝐴 = 𝐵𝐺 ∈ (V × V))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wcel 2106  Vcvv 3478  {csn 4631  cop 4637   × cxp 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571
This theorem is referenced by: (None)
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