Theorem fun2dmnopgexmpl 47988

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 12311	. . . . . . . 8 ⊢ 0 ≠ 1
2	1	neii 2958	. . . . . . 7 ⊢ ¬ 0 = 1
3	2	intnanr 492	. . . . . 6 ⊢ ¬ (0 = 1 ∧ 𝑎 = {0})
4	3	intnanr 492	. . . . 5 ⊢ ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))
5	4	gen2 1824	. . . 4 ⊢ ∀𝑎∀𝑏 ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))
6		eqeq1 2765	. . . . . . 7 ⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (𝐺 = ⟨𝑎, 𝑏⟩ ↔ {⟨0, 1⟩, ⟨1, 1⟩} = ⟨𝑎, 𝑏⟩))
7		c0ex 11199	. . . . . . . 8 ⊢ 0 ∈ V
8		1ex 11202	. . . . . . . 8 ⊢ 1 ∈ V
9		vex 3457	. . . . . . . 8 ⊢ 𝑎 ∈ V
10		vex 3457	. . . . . . . 8 ⊢ 𝑏 ∈ V
11	7, 8, 8, 8, 9, 10	propeqop 5490	. . . . . . 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 1951	. . . 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 1858	. . 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 5736	. 2 ⊢ (𝐺 ∈ (V × V) ↔ ∃𝑎∃𝑏 𝐺 = ⟨𝑎, 𝑏⟩)
19	17, 18	sylnibr 332	1 ⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ 𝐺 ∈ (V × V))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860  ∀wal 1566   = wceq 1568  ∃wex 1807   ∈ wcel 2141  Vcvv 3453  {csn 4588  {cpr 4590  ⟨cop 4594   × cxp 5659  0cc0 11099  1c1 11100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-mulcl 11161  ax-i2m1 11167  ax-1ne0 11168

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-opab 5173  df-xp 5667

This theorem is referenced by: (None)