Theorem fun2dmnopgexmpl 46292

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 12289	. . . . . . . 8 ⊢ 0 ≠ 1
2	1	neii 2940	. . . . . . 7 ⊢ ¬ 0 = 1
3	2	intnanr 486	. . . . . 6 ⊢ ¬ (0 = 1 ∧ 𝑎 = {0})
4	3	intnanr 486	. . . . 5 ⊢ ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))
5	4	gen2 1796	. . . 4 ⊢ ∀𝑎∀𝑏 ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))
6		eqeq1 2734	. . . . . . 7 ⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (𝐺 = ⟨𝑎, 𝑏⟩ ↔ {⟨0, 1⟩, ⟨1, 1⟩} = ⟨𝑎, 𝑏⟩))
7		c0ex 11214	. . . . . . . 8 ⊢ 0 ∈ V
8		1ex 11216	. . . . . . . 8 ⊢ 1 ∈ V
9		vex 3476	. . . . . . . 8 ⊢ 𝑎 ∈ V
10		vex 3476	. . . . . . . 8 ⊢ 𝑏 ∈ V
11	7, 8, 8, 8, 9, 10	propeqop 5508	. . . . . . 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 1924	. . . 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 1830	. . 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 5751	. 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 394   ∨ wo 843  ∀wal 1537   = wceq 1539  ∃wex 1779   ∈ wcel 2104  Vcvv 3472  {csn 4629  {cpr 4631  ⟨cop 4635   × cxp 5675  0cc0 11114  1c1 11115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-mulcl 11176  ax-i2m1 11182  ax-1ne0 11183

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-xp 5683

This theorem is referenced by: (None)