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Theorem funsneqopb 6324
 Description: A singleton of an ordered pair is an ordered pair iff the components are equal. (Contributed by AV, 24-Sep-2020.)
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.a . . 3 𝐴 ∈ V
2 funsndifnop.b . . 3 𝐵 ∈ V
3 funsndifnop.g . . 3 𝐺 = {⟨𝐴, 𝐵⟩}
41, 2, 3funsneqop 6323 . 2 (𝐴 = 𝐵𝐺 ∈ (V × V))
51, 2, 3funsndifnop 6321 . . 3 (𝐴𝐵 → ¬ 𝐺 ∈ (V × V))
65necon4ai 2813 . 2 (𝐺 ∈ (V × V) → 𝐴 = 𝐵)
74, 6impbii 198 1 (𝐴 = 𝐵𝐺 ∈ (V × V))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ⟨cop 4131   × cxp 5036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812 This theorem is referenced by: (None)
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