Theorem fun2dmnopgexmpl 45506

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 12224	. . . . . . . 8 ⊢ 0 ≠ 1
2	1	neii 2945	. . . . . . 7 ⊢ ¬ 0 = 1
3	2	intnanr 488	. . . . . 6 ⊢ ¬ (0 = 1 ∧ 𝑎 = {0})
4	3	intnanr 488	. . . . 5 ⊢ ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))
5	4	gen2 1798	. . . 4 ⊢ ∀𝑎∀𝑏 ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))
6		eqeq1 2740	. . . . . . 7 ⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (𝐺 = ⟨𝑎, 𝑏⟩ ↔ {⟨0, 1⟩, ⟨1, 1⟩} = ⟨𝑎, 𝑏⟩))
7		c0ex 11149	. . . . . . . 8 ⊢ 0 ∈ V
8		1ex 11151	. . . . . . . 8 ⊢ 1 ∈ V
9		vex 3449	. . . . . . . 8 ⊢ 𝑎 ∈ V
10		vex 3449	. . . . . . . 8 ⊢ 𝑏 ∈ V
11	7, 8, 8, 8, 9, 10	propeqop 5464	. . . . . . 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 286	. . . . . 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 317	. . . . 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 1926	. . . 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 257	. . 3 ⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ∀𝑎∀𝑏 ¬ 𝐺 = ⟨𝑎, 𝑏⟩)
16		2nexaln 1832	. . 3 ⊢ (¬ ∃𝑎∃𝑏 𝐺 = ⟨𝑎, 𝑏⟩ ↔ ∀𝑎∀𝑏 ¬ 𝐺 = ⟨𝑎, 𝑏⟩)
17	15, 16	sylibr 233	. 2 ⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ ∃𝑎∃𝑏 𝐺 = ⟨𝑎, 𝑏⟩)
18		elvv 5706	. 2 ⊢ (𝐺 ∈ (V × V) ↔ ∃𝑎∃𝑏 𝐺 = ⟨𝑎, 𝑏⟩)
19	17, 18	sylnibr 328	1 ⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ 𝐺 ∈ (V × V))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 845  ∀wal 1539   = wceq 1541  ∃wex 1781   ∈ wcel 2106  Vcvv 3445  {csn 4586  {cpr 4588  ⟨cop 4592   × cxp 5631  0cc0 11051  1c1 11052

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-mulcl 11113  ax-i2m1 11119  ax-1ne0 11120

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-opab 5168  df-xp 5639

This theorem is referenced by: (None)