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Theorem funsndifnop 6373
Description: A singleton of an ordered pair is not an ordered pair if the components are different. (Contributed by AV, 23-Sep-2020.)
Hypotheses
Ref Expression
funsndifnop.a 𝐴 ∈ V
funsndifnop.b 𝐵 ∈ V
funsndifnop.g 𝐺 = {⟨𝐴, 𝐵⟩}
Assertion
Ref Expression
funsndifnop (𝐴𝐵 → ¬ 𝐺 ∈ (V × V))

Proof of Theorem funsndifnop
Dummy variables 𝑎 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elvv 5140 . . 3 (𝐺 ∈ (V × V) ↔ ∃𝑥𝑦 𝐺 = ⟨𝑥, 𝑦⟩)
2 funsndifnop.g . . . . . 6 𝐺 = {⟨𝐴, 𝐵⟩}
3 funsndifnop.a . . . . . . . 8 𝐴 ∈ V
4 funsndifnop.b . . . . . . . 8 𝐵 ∈ V
53, 4funsn 5899 . . . . . . 7 Fun {⟨𝐴, 𝐵⟩}
6 funeq 5869 . . . . . . 7 (𝐺 = {⟨𝐴, 𝐵⟩} → (Fun 𝐺 ↔ Fun {⟨𝐴, 𝐵⟩}))
75, 6mpbiri 248 . . . . . 6 (𝐺 = {⟨𝐴, 𝐵⟩} → Fun 𝐺)
82, 7ax-mp 5 . . . . 5 Fun 𝐺
9 funeq 5869 . . . . . . 7 (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun 𝐺 ↔ Fun ⟨𝑥, 𝑦⟩))
10 vex 3189 . . . . . . . 8 𝑥 ∈ V
11 vex 3189 . . . . . . . 8 𝑦 ∈ V
1210, 11funop 6371 . . . . . . 7 (Fun ⟨𝑥, 𝑦⟩ ↔ ∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}))
139, 12syl6bb 276 . . . . . 6 (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun 𝐺 ↔ ∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩})))
14 eqeq2 2632 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩} → (𝐺 = ⟨𝑥, 𝑦⟩ ↔ 𝐺 = {⟨𝑎, 𝑎⟩}))
15 eqeq1 2625 . . . . . . . . . . . . 13 (𝐺 = {⟨𝐴, 𝐵⟩} → (𝐺 = {⟨𝑎, 𝑎⟩} ↔ {⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩}))
16 opex 4895 . . . . . . . . . . . . . . 15 𝐴, 𝐵⟩ ∈ V
1716sneqr 4341 . . . . . . . . . . . . . 14 ({⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩} → ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩)
183, 4opth 4907 . . . . . . . . . . . . . . 15 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩ ↔ (𝐴 = 𝑎𝐵 = 𝑎))
19 eqtr3 2642 . . . . . . . . . . . . . . . 16 ((𝐴 = 𝑎𝐵 = 𝑎) → 𝐴 = 𝐵)
2019a1d 25 . . . . . . . . . . . . . . 15 ((𝐴 = 𝑎𝐵 = 𝑎) → (𝑥 = {𝑎} → 𝐴 = 𝐵))
2118, 20sylbi 207 . . . . . . . . . . . . . 14 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩ → (𝑥 = {𝑎} → 𝐴 = 𝐵))
2217, 21syl 17 . . . . . . . . . . . . 13 ({⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩} → (𝑥 = {𝑎} → 𝐴 = 𝐵))
2315, 22syl6bi 243 . . . . . . . . . . . 12 (𝐺 = {⟨𝐴, 𝐵⟩} → (𝐺 = {⟨𝑎, 𝑎⟩} → (𝑥 = {𝑎} → 𝐴 = 𝐵)))
242, 23ax-mp 5 . . . . . . . . . . 11 (𝐺 = {⟨𝑎, 𝑎⟩} → (𝑥 = {𝑎} → 𝐴 = 𝐵))
2514, 24syl6bi 243 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩} → (𝐺 = ⟨𝑥, 𝑦⟩ → (𝑥 = {𝑎} → 𝐴 = 𝐵)))
2625com23 86 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩} → (𝑥 = {𝑎} → (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵)))
2726impcom 446 . . . . . . . 8 ((𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}) → (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵))
2827exlimiv 1855 . . . . . . 7 (∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}) → (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵))
2928com12 32 . . . . . 6 (𝐺 = ⟨𝑥, 𝑦⟩ → (∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}) → 𝐴 = 𝐵))
3013, 29sylbid 230 . . . . 5 (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun 𝐺𝐴 = 𝐵))
318, 30mpi 20 . . . 4 (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵)
3231exlimivv 1857 . . 3 (∃𝑥𝑦 𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵)
331, 32sylbi 207 . 2 (𝐺 ∈ (V × V) → 𝐴 = 𝐵)
3433necon3ai 2815 1 (𝐴𝐵 → ¬ 𝐺 ∈ (V × V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  Vcvv 3186  {csn 4150  cop 4156   × cxp 5074  Fun wfun 5843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857
This theorem is referenced by:  funsneqopb  6376  snstrvtxval  25836  snstriedgval  25837
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