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Theorem fun2dmnopgexmpl 42135
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 11384 . . . . . . . 8 0 ≠ 1
21neii 2973 . . . . . . 7 ¬ 0 = 1
32intnanr 482 . . . . . 6 ¬ (0 = 1 ∧ 𝑎 = {0})
43intnanr 482 . . . . 5 ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))
54gen2 1892 . . . 4 𝑎𝑏 ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))
6 eqeq1 2803 . . . . . . . 8 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (𝐺 = ⟨𝑎, 𝑏⟩ ↔ {⟨0, 1⟩, ⟨1, 1⟩} = ⟨𝑎, 𝑏⟩))
7 c0ex 10322 . . . . . . . . 9 0 ∈ V
8 1ex 10324 . . . . . . . . 9 1 ∈ V
9 vex 3388 . . . . . . . . 9 𝑎 ∈ V
10 vex 3388 . . . . . . . . 9 𝑏 ∈ V
117, 8, 8, 8, 9, 10propeqop 5163 . . . . . . . 8 ({⟨0, 1⟩, ⟨1, 1⟩} = ⟨𝑎, 𝑏⟩ ↔ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1}))))
126, 11syl6bb 279 . . . . . . 7 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (𝐺 = ⟨𝑎, 𝑏⟩ ↔ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))))
1312notbid 310 . . . . . 6 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (¬ 𝐺 = ⟨𝑎, 𝑏⟩ ↔ ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))))
1413albidv 2016 . . . . 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 2016 . . . 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 250 . . 3 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ∀𝑎𝑏 ¬ 𝐺 = ⟨𝑎, 𝑏⟩)
17 2nexaln 1925 . . 3 (¬ ∃𝑎𝑏 𝐺 = ⟨𝑎, 𝑏⟩ ↔ ∀𝑎𝑏 ¬ 𝐺 = ⟨𝑎, 𝑏⟩)
1816, 17sylibr 226 . 2 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ ∃𝑎𝑏 𝐺 = ⟨𝑎, 𝑏⟩)
19 elvv 5380 . 2 (𝐺 ∈ (V × V) ↔ ∃𝑎𝑏 𝐺 = ⟨𝑎, 𝑏⟩)
2018, 19sylnibr 321 1 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ 𝐺 ∈ (V × V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385  wo 874  wal 1651   = wceq 1653  wex 1875  wcel 2157  Vcvv 3385  {csn 4368  {cpr 4370  cop 4374   × cxp 5310  0cc0 10224  1c1 10225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-mulcl 10286  ax-i2m1 10292  ax-1ne0 10293
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-opab 4906  df-xp 5318
This theorem is referenced by: (None)
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