Theorem prneimg 4818

Description: Two pairs are not equal if at least one element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.)

Assertion

Ref	Expression
prneimg	⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → (((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}))

Proof of Theorem prneimg

Step	Hyp	Ref	Expression
1		preq12bg 4817	. . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
2		orddi 1026	. . . . . 6 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) ↔ (((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐴 = 𝐶 ∨ 𝐵 = 𝐶)) ∧ ((𝐵 = 𝐷 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐷 ∨ 𝐵 = 𝐶))))
3		simpll 778	. . . . . . 7 ⊢ ((((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐴 = 𝐶 ∨ 𝐵 = 𝐶)) ∧ ((𝐵 = 𝐷 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐷 ∨ 𝐵 = 𝐶))) → (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
4		pm1.4 882	. . . . . . . 8 ⊢ ((𝐵 = 𝐷 ∨ 𝐵 = 𝐶) → (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))
5	4	ad2antll 741	. . . . . . 7 ⊢ ((((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐴 = 𝐶 ∨ 𝐵 = 𝐶)) ∧ ((𝐵 = 𝐷 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐷 ∨ 𝐵 = 𝐶))) → (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))
6	3, 5	jca 520	. . . . . 6 ⊢ ((((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐴 = 𝐶 ∨ 𝐵 = 𝐶)) ∧ ((𝐵 = 𝐷 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐷 ∨ 𝐵 = 𝐶))) → ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)))
7	2, 6	sylbi 220	. . . . 5 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) → ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)))
8	1, 7	biimtrdi 256	. . . 4 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))))
9		ianor 996	. . . . . 6 ⊢ (¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ↔ (¬ 𝐴 ≠ 𝐶 ∨ ¬ 𝐴 ≠ 𝐷))
10		nne 2961	. . . . . . 7 ⊢ (¬ 𝐴 ≠ 𝐶 ↔ 𝐴 = 𝐶)
11		nne 2961	. . . . . . 7 ⊢ (¬ 𝐴 ≠ 𝐷 ↔ 𝐴 = 𝐷)
12	10, 11	orbi12i 927	. . . . . 6 ⊢ ((¬ 𝐴 ≠ 𝐶 ∨ ¬ 𝐴 ≠ 𝐷) ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
13	9, 12	bitr2i 279	. . . . 5 ⊢ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ↔ ¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷))
14		ianor 996	. . . . . 6 ⊢ (¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ↔ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐵 ≠ 𝐷))
15		nne 2961	. . . . . . 7 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
16		nne 2961	. . . . . . 7 ⊢ (¬ 𝐵 ≠ 𝐷 ↔ 𝐵 = 𝐷)
17	15, 16	orbi12i 927	. . . . . 6 ⊢ ((¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐵 ≠ 𝐷) ↔ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))
18	14, 17	bitr2i 279	. . . . 5 ⊢ ((𝐵 = 𝐶 ∨ 𝐵 = 𝐷) ↔ ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷))
19	13, 18	anbi12i 639	. . . 4 ⊢ (((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)) ↔ (¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)))
20	8, 19	imbitrdi 254	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → (¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷))))
21		pm4.56 1003	. . 3 ⊢ ((¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) ↔ ¬ ((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)))
22	20, 21	imbitrdi 254	. 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ¬ ((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷))))
23	22	necon2ad 2972	1 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → (((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  {cpr 4590

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3456  df-un 3909  df-sn 4589  df-pr 4591

This theorem is used by:  prnebg  4820  opthhausdorff  5499  symg2bas  19469  m2detleib  22799  umgrvad2edg  29574  usgrexmpldifpr  29619  usgrexmpl1lem  48814  usgrexmpl2lem  48819  usgrexmpl2nb0  48824  usgrexmpl2nb1  48825  usgrexmpl2nb2  48826  usgrexmpl2nb3  48827  usgrexmpl2nb4  48828  usgrexmpl2nb5  48829  gpg5nbgrvtx03starlem1  48861  gpg5nbgrvtx03starlem2  48862  gpg5nbgrvtx03starlem3  48863  gpg5nbgrvtx13starlem1  48864  gpg5nbgrvtx13starlem2  48865  gpg5nbgrvtx13starlem3  48866  gpgprismgr4cycllem2  48889  gpg5edgnedg  48923  zlmodzxzldeplem  49306  line2x  49562  inlinecirc02plem  49594