Theorem fun2dmnopgexmpl 48559

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.

Step	Hyp	Ref	Expression
1		0ne1 12441	. . . . . . . 8 ⊢ 0 ≠ 1
2	1	neii 2961	. . . . . . 7 ⊢ ¬ 0 = 1
3	2	intnanr 493	. . . . . 6 ⊢ ¬ (0 = 1 ∧ 𝑎 = {0})
4	3	intnanr 493	. . . . 5 ⊢ ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))
5	4	gen2 1829	. . . 4 ⊢ ∀𝑎∀𝑏 ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))
6		eqeq1 2767	. . . . . . 7 ⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (𝐺 = ⟨𝑎, 𝑏⟩ ↔ {⟨0, 1⟩, ⟨1, 1⟩} = ⟨𝑎, 𝑏⟩))
7		c0ex 11327	. . . . . . . 8 ⊢ 0 ∈ V
8		1ex 11330	. . . . . . . 8 ⊢ 1 ∈ V
9		vex 3458	. . . . . . . 8 ⊢ 𝑎 ∈ V
10		vex 3458	. . . . . . . 8 ⊢ 𝑏 ∈ V
11	7, 8, 8, 8, 9, 10	propeqop 5483	. . . . . . 7 ⊢ ({⟨0, 1⟩, ⟨1, 1⟩} = ⟨𝑎, 𝑏⟩ ↔ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1}))))
12	6, 11	bitrdi 290	. . . . . 6 ⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (𝐺 = ⟨𝑎, 𝑏⟩ ↔ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))))
13	12	notbid 321	. . . . 5 ⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (¬ 𝐺 = ⟨𝑎, 𝑏⟩ ↔ ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))))
14	13	2albidv 1958	. . . 4 ⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (∀𝑎∀𝑏 ¬ 𝐺 = ⟨𝑎, 𝑏⟩ ↔ ∀𝑎∀𝑏 ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))))
15	5, 14	mpbiri 261	. . 3 ⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ∀𝑎∀𝑏 ¬ 𝐺 = ⟨𝑎, 𝑏⟩)
16		2nexaln 1863	. . 3 ⊢ (¬ ∃𝑎∃𝑏 𝐺 = ⟨𝑎, 𝑏⟩ ↔ ∀𝑎∀𝑏 ¬ 𝐺 = ⟨𝑎, 𝑏⟩)
17	15, 16	sylibr 237	. 2 ⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ ∃𝑎∃𝑏 𝐺 = ⟨𝑎, 𝑏⟩)
18		elvv 5730	. 2 ⊢ (𝐺 ∈ (V × V) ↔ ∃𝑎∃𝑏 𝐺 = ⟨𝑎, 𝑏⟩)
19	17, 18	sylnibr 332	1 ⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ 𝐺 ∈ (V × V))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454  {csn 4588  {cpr 4590  ⟨cop 4838   × cxp 5653  0cc0 11227  1c1 11228

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-mulcl 11289  ax-i2m1 11295  ax-1ne0 11296

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-opab 5171  df-xp 5661

This theorem is used by: (None)