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Theorem funsneqop 6323
 Description: A singleton of an ordered pair is an ordered pair if 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
funsneqop (𝐴 = 𝐵𝐺 ∈ (V × V))

Proof of Theorem funsneqop
StepHypRef Expression
1 funsndifnop.a . . 3 𝐴 ∈ V
2 funsndifnop.b . . 3 𝐵 ∈ V
3 funsndifnop.g . . 3 𝐺 = {⟨𝐴, 𝐵⟩}
41, 2, 3funsneqopsn 6322 . 2 (𝐴 = 𝐵𝐺 = ⟨{𝐴}, {𝐴}⟩)
5 snex 4835 . . 3 {𝐴} ∈ V
65, 5opelvv 5088 . 2 ⟨{𝐴}, {𝐴}⟩ ∈ (V × V)
74, 6syl6eqel 2696 1 (𝐴 = 𝐵𝐺 ∈ (V × V))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = 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-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  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-opab 4644  df-xp 5044 This theorem is referenced by:  funsneqopb  6324
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