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Theorem funsndifnop 6608
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 5345 . . 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 6120 . . . . . . 7 Fun {⟨𝐴, 𝐵⟩}
6 funeq 6088 . . . . . . 7 (𝐺 = {⟨𝐴, 𝐵⟩} → (Fun 𝐺 ↔ Fun {⟨𝐴, 𝐵⟩}))
75, 6mpbiri 249 . . . . . 6 (𝐺 = {⟨𝐴, 𝐵⟩} → Fun 𝐺)
82, 7ax-mp 5 . . . . 5 Fun 𝐺
9 funeq 6088 . . . . . . 7 (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun 𝐺 ↔ Fun ⟨𝑥, 𝑦⟩))
10 vex 3353 . . . . . . . 8 𝑥 ∈ V
11 vex 3353 . . . . . . . 8 𝑦 ∈ V
1210, 11funop 6606 . . . . . . 7 (Fun ⟨𝑥, 𝑦⟩ ↔ ∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}))
139, 12syl6bb 278 . . . . . 6 (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun 𝐺 ↔ ∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩})))
14 eqeq2 2776 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩} → (𝐺 = ⟨𝑥, 𝑦⟩ ↔ 𝐺 = {⟨𝑎, 𝑎⟩}))
15 eqeq1 2769 . . . . . . . . . . . . 13 (𝐺 = {⟨𝐴, 𝐵⟩} → (𝐺 = {⟨𝑎, 𝑎⟩} ↔ {⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩}))
16 opex 5088 . . . . . . . . . . . . . . 15 𝐴, 𝐵⟩ ∈ V
1716sneqr 4523 . . . . . . . . . . . . . 14 ({⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩} → ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩)
183, 4opth 5100 . . . . . . . . . . . . . . 15 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩ ↔ (𝐴 = 𝑎𝐵 = 𝑎))
19 eqtr3 2786 . . . . . . . . . . . . . . . 16 ((𝐴 = 𝑎𝐵 = 𝑎) → 𝐴 = 𝐵)
2019a1d 25 . . . . . . . . . . . . . . 15 ((𝐴 = 𝑎𝐵 = 𝑎) → (𝑥 = {𝑎} → 𝐴 = 𝐵))
2118, 20sylbi 208 . . . . . . . . . . . . . 14 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩ → (𝑥 = {𝑎} → 𝐴 = 𝐵))
2217, 21syl 17 . . . . . . . . . . . . 13 ({⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩} → (𝑥 = {𝑎} → 𝐴 = 𝐵))
2315, 22syl6bi 244 . . . . . . . . . . . 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 244 . . . . . . . . . 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 396 . . . . . . . 8 ((𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}) → (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵))
2827exlimiv 2025 . . . . . . 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 231 . . . . 5 (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun 𝐺𝐴 = 𝐵))
318, 30mpi 20 . . . 4 (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵)
3231exlimivv 2027 . . 3 (∃𝑥𝑦 𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵)
331, 32sylbi 208 . 2 (𝐺 ∈ (V × V) → 𝐴 = 𝐵)
3433necon3ai 2962 1 (𝐴𝐵 → ¬ 𝐺 ∈ (V × V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1652  wex 1874  wcel 2155  wne 2937  Vcvv 3350  {csn 4334  cop 4340   × cxp 5275  Fun wfun 6062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076
This theorem is referenced by:  funsneqopb  6611  snstrvtxval  26206  snstriedgval  26207
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