Theorem fun2dmnopgexmpl 40329

Description: A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.)

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 10965	. . . . . . . 8 ⊢ 0 ≠ 1
2	1	neii 2784	. . . . . . 7 ⊢ ¬ 0 = 1
3	2	intnanr 952	. . . . . 6 ⊢ ¬ (0 = 1 ∧ 𝑎 = {0})
4	3	intnanr 952	. . . . 5 ⊢ ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))
5	4	gen2 1714	. . . 4 ⊢ ∀𝑎∀𝑏 ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))
6		eqeq1 2614	. . . . . . . 8 ⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (𝐺 = ⟨𝑎, 𝑏⟩ ↔ {⟨0, 1⟩, ⟨1, 1⟩} = ⟨𝑎, 𝑏⟩))
7		c0ex 9913	. . . . . . . . 9 ⊢ 0 ∈ V
8		1ex 9914	. . . . . . . . 9 ⊢ 1 ∈ V
9		vex 3176	. . . . . . . . 9 ⊢ 𝑎 ∈ V
10		vex 3176	. . . . . . . . 9 ⊢ 𝑏 ∈ V
11	7, 8, 8, 8, 9, 10	propeqop 4895	. . . . . . . 8 ⊢ ({⟨0, 1⟩, ⟨1, 1⟩} = ⟨𝑎, 𝑏⟩ ↔ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1}))))
12	6, 11	syl6bb 275	. . . . . . 7 ⊢ (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 307	. . . . . 6 ⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → (¬ 𝐺 = ⟨𝑎, 𝑏⟩ ↔ ¬ ((0 = 1 ∧ 𝑎 = {0}) ∧ ((0 = 1 ∧ 𝑏 = {0, 1}) ∨ (0 = 1 ∧ 𝑏 = {0, 1})))))
14	13	albidv 1836	. . . . 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})))))
15	14	albidv 1836	. . . 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})))))
16	5, 15	mpbiri 247	. . 3 ⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ∀𝑎∀𝑏 ¬ 𝐺 = ⟨𝑎, 𝑏⟩)
17		2nexaln 1747	. . 3 ⊢ (¬ ∃𝑎∃𝑏 𝐺 = ⟨𝑎, 𝑏⟩ ↔ ∀𝑎∀𝑏 ¬ 𝐺 = ⟨𝑎, 𝑏⟩)
18	16, 17	sylibr 223	. 2 ⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ ∃𝑎∃𝑏 𝐺 = ⟨𝑎, 𝑏⟩)
19		elvv 5100	. 2 ⊢ (𝐺 ∈ (V × V) ↔ ∃𝑎∃𝑏 𝐺 = ⟨𝑎, 𝑏⟩)
20	18, 19	sylnibr 318	1 ⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ 𝐺 ∈ (V × V))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  {csn 4125  {cpr 4127  ⟨cop 4131   × cxp 5036  0cc0 9815  1c1 9816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-mulcl 9877  ax-i2m1 9883  ax-1ne0 9884

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044

This theorem is referenced by: (None)