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Theorem funsneqopb 6906
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 4795 . . . . . 6 (𝐴 = 𝐵 → ⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐵⟩)
32sneqd 4569 . . . . 5 (𝐴 = 𝐵 → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐵⟩})
4 funsndifnop.b . . . . . 6 𝐵 ∈ V
54snopeqopsnid 5390 . . . . 5 {⟨𝐵, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩
63, 5syl6eq 2869 . . . 4 (𝐴 = 𝐵 → {⟨𝐴, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩)
71, 6syl5eq 2865 . . 3 (𝐴 = 𝐵𝐺 = ⟨{𝐵}, {𝐵}⟩)
8 snex 5322 . . . 4 {𝐵} ∈ V
98, 8opelvv 5587 . . 3 ⟨{𝐵}, {𝐵}⟩ ∈ (V × V)
107, 9syl6eqel 2918 . 2 (𝐴 = 𝐵𝐺 ∈ (V × V))
11 funsndifnop.a . . . 4 𝐴 ∈ V
1211, 4, 1funsndifnop 6905 . . 3 (𝐴𝐵 → ¬ 𝐺 ∈ (V × V))
1312necon4ai 3044 . 2 (𝐺 ∈ (V × V) → 𝐴 = 𝐵)
1410, 13impbii 210 1 (𝐴 = 𝐵𝐺 ∈ (V × V))
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wcel 2105  Vcvv 3492  {csn 4557  cop 4563   × cxp 5546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356
This theorem is referenced by: (None)
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