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Theorem funsneqopb 6901
 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 4767 . . . . . 6 (𝐴 = 𝐵 → ⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐵⟩)
32sneqd 4540 . . . . 5 (𝐴 = 𝐵 → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐵⟩})
4 funsndifnop.b . . . . . 6 𝐵 ∈ V
54snopeqopsnid 5368 . . . . 5 {⟨𝐵, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩
63, 5eqtrdi 2849 . . . 4 (𝐴 = 𝐵 → {⟨𝐴, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩)
71, 6syl5eq 2845 . . 3 (𝐴 = 𝐵𝐺 = ⟨{𝐵}, {𝐵}⟩)
8 snex 5301 . . . 4 {𝐵} ∈ V
98, 8opelvv 5562 . . 3 ⟨{𝐵}, {𝐵}⟩ ∈ (V × V)
107, 9eqeltrdi 2898 . 2 (𝐴 = 𝐵𝐺 ∈ (V × V))
11 funsndifnop.a . . . 4 𝐴 ∈ V
1211, 4, 1funsndifnop 6900 . . 3 (𝐴𝐵 → ¬ 𝐺 ∈ (V × V))
1312necon4ai 3018 . 2 (𝐺 ∈ (V × V) → 𝐴 = 𝐵)
1410, 13impbii 212 1 (𝐴 = 𝐵𝐺 ∈ (V × V))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  Vcvv 3442  {csn 4528  ⟨cop 4534   × cxp 5521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340 This theorem is referenced by: (None)
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