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Theorem fun2dmnopgexmpl 41823
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 11294 . . . . . . . 8 0 ≠ 1
21neii 2945 . . . . . . 7 ¬ 0 = 1
32intnanr 475 . . . . . 6 ¬ (0 = 1 ∧ 𝑎 = {0})
43intnanr 475 . . . . 5 ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))
54gen2 1871 . . . 4 𝑎𝑏 ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))
6 eqeq1 2775 . . . . . . . 8 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (𝐺 = ⟨𝑎, 𝑏⟩ ↔ {⟨0, 1⟩, ⟨1, 1⟩} = ⟨𝑎, 𝑏⟩))
7 c0ex 10240 . . . . . . . . 9 0 ∈ V
8 1ex 10241 . . . . . . . . 9 1 ∈ V
9 vex 3354 . . . . . . . . 9 𝑎 ∈ V
10 vex 3354 . . . . . . . . 9 𝑏 ∈ V
117, 8, 8, 8, 9, 10propeqop 5101 . . . . . . . 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 307 . . . . . 6 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (¬ 𝐺 = ⟨𝑎, 𝑏⟩ ↔ ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))))
1413albidv 2001 . . . . 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 2001 . . . 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 1905 . . 3 (¬ ∃𝑎𝑏 𝐺 = ⟨𝑎, 𝑏⟩ ↔ ∀𝑎𝑏 ¬ 𝐺 = ⟨𝑎, 𝑏⟩)
1816, 17sylibr 224 . 2 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ ∃𝑎𝑏 𝐺 = ⟨𝑎, 𝑏⟩)
19 elvv 5316 . 2 (𝐺 ∈ (V × V) ↔ ∃𝑎𝑏 𝐺 = ⟨𝑎, 𝑏⟩)
2018, 19sylnibr 318 1 (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ 𝐺 ∈ (V × V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wo 836  wal 1629   = wceq 1631  wex 1852  wcel 2145  Vcvv 3351  {csn 4317  {cpr 4319  cop 4323   × cxp 5248  0cc0 10142  1c1 10143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-mulcl 10204  ax-i2m1 10210  ax-1ne0 10211
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-opab 4848  df-xp 5256
This theorem is referenced by: (None)
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