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Theorem funsndifnop 6559
Description: A singleton of an ordered pair is not an ordered pair if the components are different. (Contributed by AV, 23-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
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 5317 . . 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 6082 . . . . . . 7 Fun {⟨𝐴, 𝐵⟩}
6 funeq 6051 . . . . . . 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 6051 . . . . . . 7 (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun 𝐺 ↔ Fun ⟨𝑥, 𝑦⟩))
10 vex 3354 . . . . . . . 8 𝑥 ∈ V
11 vex 3354 . . . . . . . 8 𝑦 ∈ V
1210, 11funop 6557 . . . . . . 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 2782 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩} → (𝐺 = ⟨𝑥, 𝑦⟩ ↔ 𝐺 = {⟨𝑎, 𝑎⟩}))
15 eqeq1 2775 . . . . . . . . . . . . 13 (𝐺 = {⟨𝐴, 𝐵⟩} → (𝐺 = {⟨𝑎, 𝑎⟩} ↔ {⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩}))
16 opex 5060 . . . . . . . . . . . . . . 15 𝐴, 𝐵⟩ ∈ V
1716sneqr 4504 . . . . . . . . . . . . . 14 ({⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩} → ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩)
183, 4opth 5072 . . . . . . . . . . . . . . 15 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩ ↔ (𝐴 = 𝑎𝐵 = 𝑎))
19 eqtr3 2792 . . . . . . . . . . . . . . . 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 394 . . . . . . . 8 ((𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}) → (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵))
2827exlimiv 2010 . . . . . . 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 2012 . . 3 (∃𝑥𝑦 𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵)
331, 32sylbi 207 . 2 (𝐺 ∈ (V × V) → 𝐴 = 𝐵)
3433necon3ai 2968 1 (𝐴𝐵 → ¬ 𝐺 ∈ (V × V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wex 1852  wcel 2145  wne 2943  Vcvv 3351  {csn 4316  cop 4322   × cxp 5247  Fun wfun 6025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039
This theorem is referenced by:  funsneqopb  6562  snstrvtxval  26150  snstriedgval  26151
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