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Theorem fun2dmnopgexmpl 40600
Description: A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.)
Assertion
Ref Expression
fun2dmnopgexmpl (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ 𝐺 ∈ (V × V))

Proof of Theorem fun2dmnopgexmpl
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ne1 11032 . . . . . . . 8 0 ≠ 1
21neii 2792 . . . . . . 7 ¬ 0 = 1
32intnanr 960 . . . . . 6 ¬ (0 = 1 ∧ 𝑎 = {0})
43intnanr 960 . . . . 5 ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))
54gen2 1720 . . . 4 𝑎𝑏 ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))
6 eqeq1 2625 . . . . . . . 8 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (𝐺 = ⟨𝑎, 𝑏⟩ ↔ {⟨0, 1⟩, ⟨1, 1⟩} = ⟨𝑎, 𝑏⟩))
7 c0ex 9978 . . . . . . . . 9 0 ∈ V
8 1ex 9979 . . . . . . . . 9 1 ∈ V
9 vex 3189 . . . . . . . . 9 𝑎 ∈ V
10 vex 3189 . . . . . . . . 9 𝑏 ∈ V
117, 8, 8, 8, 9, 10propeqop 4930 . . . . . . . 8 ({⟨0, 1⟩, ⟨1, 1⟩} = ⟨𝑎, 𝑏⟩ ↔ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1}))))
126, 11syl6bb 276 . . . . . . 7 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (𝐺 = ⟨𝑎, 𝑏⟩ ↔ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))))
1312notbid 308 . . . . . 6 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (¬ 𝐺 = ⟨𝑎, 𝑏⟩ ↔ ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))))
1413albidv 1846 . . . . 5 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (∀𝑏 ¬ 𝐺 = ⟨𝑎, 𝑏⟩ ↔ ∀𝑏 ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))))
1514albidv 1846 . . . 4 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (∀𝑎𝑏 ¬ 𝐺 = ⟨𝑎, 𝑏⟩ ↔ ∀𝑎𝑏 ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))))
165, 15mpbiri 248 . . 3 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ∀𝑎𝑏 ¬ 𝐺 = ⟨𝑎, 𝑏⟩)
17 2nexaln 1754 . . 3 (¬ ∃𝑎𝑏 𝐺 = ⟨𝑎, 𝑏⟩ ↔ ∀𝑎𝑏 ¬ 𝐺 = ⟨𝑎, 𝑏⟩)
1816, 17sylibr 224 . 2 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ ∃𝑎𝑏 𝐺 = ⟨𝑎, 𝑏⟩)
19 elvv 5138 . 2 (𝐺 ∈ (V × V) ↔ ∃𝑎𝑏 𝐺 = ⟨𝑎, 𝑏⟩)
2018, 19sylnibr 319 1 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ 𝐺 ∈ (V × V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186  {csn 4148  {cpr 4150  cop 4154   × cxp 5072  0cc0 9880  1c1 9881
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 4741  ax-nul 4749  ax-pr 4867  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-mulcl 9942  ax-i2m1 9948  ax-1ne0 9949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-opab 4674  df-xp 5080
This theorem is referenced by: (None)
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