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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 4873 . . . . . 6 (𝐴 = 𝐵 → ⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐵⟩)
32sneqd 4638 . . . . 5 (𝐴 = 𝐵 → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐵⟩})
4 funsndifnop.b . . . . . 6 𝐵 ∈ V
54snopeqopsnid 5514 . . . . 5 {⟨𝐵, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩
63, 5eqtrdi 2793 . . . 4 (𝐴 = 𝐵 → {⟨𝐴, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩)
71, 6eqtrid 2789 . . 3 (𝐴 = 𝐵𝐺 = ⟨{𝐵}, {𝐵}⟩)
8 snex 5436 . . . 4 {𝐵} ∈ V
98, 8opelvv 5725 . . 3 ⟨{𝐵}, {𝐵}⟩ ∈ (V × V)
107, 9eqeltrdi 2849 . 2 (𝐴 = 𝐵𝐺 ∈ (V × V))
11 funsndifnop.a . . . 4 𝐴 ∈ V
1211, 4, 1funsndifnop 7171 . . 3 (𝐴𝐵 → ¬ 𝐺 ∈ (V × V))
1312necon4ai 2972 . 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 1540  wcel 2108  Vcvv 3480  {csn 4626  cop 4632   × cxp 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by: (None)
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