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Theorem funsndifnop 7171
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 5763 . . 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 6621 . . . . . . 7 Fun {⟨𝐴, 𝐵⟩}
6 funeq 6588 . . . . . . 7 (𝐺 = {⟨𝐴, 𝐵⟩} → (Fun 𝐺 ↔ Fun {⟨𝐴, 𝐵⟩}))
75, 6mpbiri 258 . . . . . 6 (𝐺 = {⟨𝐴, 𝐵⟩} → Fun 𝐺)
82, 7ax-mp 5 . . . . 5 Fun 𝐺
9 funeq 6588 . . . . . . 7 (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun 𝐺 ↔ Fun ⟨𝑥, 𝑦⟩))
10 vex 3482 . . . . . . . 8 𝑥 ∈ V
11 vex 3482 . . . . . . . 8 𝑦 ∈ V
1210, 11funop 7169 . . . . . . 7 (Fun ⟨𝑥, 𝑦⟩ ↔ ∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}))
139, 12bitrdi 287 . . . . . 6 (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun 𝐺 ↔ ∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩})))
14 eqeq2 2747 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩} → (𝐺 = ⟨𝑥, 𝑦⟩ ↔ 𝐺 = {⟨𝑎, 𝑎⟩}))
15 eqeq1 2739 . . . . . . . . . . . . 13 (𝐺 = {⟨𝐴, 𝐵⟩} → (𝐺 = {⟨𝑎, 𝑎⟩} ↔ {⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩}))
16 opex 5475 . . . . . . . . . . . . . . 15 𝐴, 𝐵⟩ ∈ V
1716sneqr 4845 . . . . . . . . . . . . . 14 ({⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩} → ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩)
183, 4opth 5487 . . . . . . . . . . . . . . 15 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩ ↔ (𝐴 = 𝑎𝐵 = 𝑎))
19 eqtr3 2761 . . . . . . . . . . . . . . . 16 ((𝐴 = 𝑎𝐵 = 𝑎) → 𝐴 = 𝐵)
2019a1d 25 . . . . . . . . . . . . . . 15 ((𝐴 = 𝑎𝐵 = 𝑎) → (𝑥 = {𝑎} → 𝐴 = 𝐵))
2118, 20sylbi 217 . . . . . . . . . . . . . 14 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩ → (𝑥 = {𝑎} → 𝐴 = 𝐵))
2217, 21syl 17 . . . . . . . . . . . . 13 ({⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩} → (𝑥 = {𝑎} → 𝐴 = 𝐵))
2315, 22biimtrdi 253 . . . . . . . . . . . 12 (𝐺 = {⟨𝐴, 𝐵⟩} → (𝐺 = {⟨𝑎, 𝑎⟩} → (𝑥 = {𝑎} → 𝐴 = 𝐵)))
242, 23ax-mp 5 . . . . . . . . . . 11 (𝐺 = {⟨𝑎, 𝑎⟩} → (𝑥 = {𝑎} → 𝐴 = 𝐵))
2514, 24biimtrdi 253 . . . . . . . . . 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 407 . . . . . . . 8 ((𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}) → (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵))
2827exlimiv 1928 . . . . . . 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 240 . . . . 5 (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun 𝐺𝐴 = 𝐵))
318, 30mpi 20 . . . 4 (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵)
3231exlimivv 1930 . . 3 (∃𝑥𝑦 𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵)
331, 32sylbi 217 . 2 (𝐺 ∈ (V × V) → 𝐴 = 𝐵)
3433necon3ai 2963 1 (𝐴𝐵 → ¬ 𝐺 ∈ (V × V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  wne 2938  Vcvv 3478  {csn 4631  cop 4637   × cxp 5687  Fun wfun 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571
This theorem is referenced by:  funsneqopb  7172  snstrvtxval  29069  snstriedgval  29070
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