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Theorem fun2dmnopgexmpl 43924
 Description: A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.) (Avoid depending on this detail.)
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 11711 . . . . . . . 8 0 ≠ 1
21neii 2989 . . . . . . 7 ¬ 0 = 1
32intnanr 491 . . . . . 6 ¬ (0 = 1 ∧ 𝑎 = {0})
43intnanr 491 . . . . 5 ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))
54gen2 1798 . . . 4 𝑎𝑏 ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))
6 eqeq1 2802 . . . . . . 7 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (𝐺 = ⟨𝑎, 𝑏⟩ ↔ {⟨0, 1⟩, ⟨1, 1⟩} = ⟨𝑎, 𝑏⟩))
7 c0ex 10639 . . . . . . . 8 0 ∈ V
8 1ex 10641 . . . . . . . 8 1 ∈ V
9 vex 3444 . . . . . . . 8 𝑎 ∈ V
10 vex 3444 . . . . . . . 8 𝑏 ∈ V
117, 8, 8, 8, 9, 10propeqop 5365 . . . . . . 7 ({⟨0, 1⟩, ⟨1, 1⟩} = ⟨𝑎, 𝑏⟩ ↔ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1}))))
126, 11syl6bb 290 . . . . . 6 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (𝐺 = ⟨𝑎, 𝑏⟩ ↔ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))))
1312notbid 321 . . . . 5 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (¬ 𝐺 = ⟨𝑎, 𝑏⟩ ↔ ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))))
14132albidv 1924 . . . 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})))))
155, 14mpbiri 261 . . 3 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ∀𝑎𝑏 ¬ 𝐺 = ⟨𝑎, 𝑏⟩)
16 2nexaln 1831 . . 3 (¬ ∃𝑎𝑏 𝐺 = ⟨𝑎, 𝑏⟩ ↔ ∀𝑎𝑏 ¬ 𝐺 = ⟨𝑎, 𝑏⟩)
1715, 16sylibr 237 . 2 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ ∃𝑎𝑏 𝐺 = ⟨𝑎, 𝑏⟩)
18 elvv 5593 . 2 (𝐺 ∈ (V × V) ↔ ∃𝑎𝑏 𝐺 = ⟨𝑎, 𝑏⟩)
1917, 18sylnibr 332 1 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ 𝐺 ∈ (V × V))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441  {csn 4527  {cpr 4529  ⟨cop 4533   × cxp 5520  0cc0 10541  1c1 10542 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-1cn 10599  ax-icn 10600  ax-addcl 10601  ax-mulcl 10603  ax-i2m1 10609  ax-1ne0 10610 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-v 3443  df-dif 3885  df-un 3887  df-nul 4246  df-if 4428  df-sn 4528  df-pr 4530  df-op 4534  df-opab 5096  df-xp 5528 This theorem is referenced by: (None)
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