Theorem prneimg 4805

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 4804	. . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
2		orddi 1011	. . . . . 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 766	. . . . . . 7 ⊢ ((((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐴 = 𝐶 ∨ 𝐵 = 𝐶)) ∧ ((𝐵 = 𝐷 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐷 ∨ 𝐵 = 𝐶))) → (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
4		pm1.4 869	. . . . . . . 8 ⊢ ((𝐵 = 𝐷 ∨ 𝐵 = 𝐶) → (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))
5	4	ad2antll 729	. . . . . . 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 511	. . . . . 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 217	. . . . 5 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) → ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)))
8	1, 7	biimtrdi 253	. . . 4 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))))
9		ianor 983	. . . . . 6 ⊢ (¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ↔ (¬ 𝐴 ≠ 𝐶 ∨ ¬ 𝐴 ≠ 𝐷))
10		nne 2929	. . . . . . 7 ⊢ (¬ 𝐴 ≠ 𝐶 ↔ 𝐴 = 𝐶)
11		nne 2929	. . . . . . 7 ⊢ (¬ 𝐴 ≠ 𝐷 ↔ 𝐴 = 𝐷)
12	10, 11	orbi12i 914	. . . . . 6 ⊢ ((¬ 𝐴 ≠ 𝐶 ∨ ¬ 𝐴 ≠ 𝐷) ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
13	9, 12	bitr2i 276	. . . . 5 ⊢ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ↔ ¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷))
14		ianor 983	. . . . . 6 ⊢ (¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ↔ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐵 ≠ 𝐷))
15		nne 2929	. . . . . . 7 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
16		nne 2929	. . . . . . 7 ⊢ (¬ 𝐵 ≠ 𝐷 ↔ 𝐵 = 𝐷)
17	15, 16	orbi12i 914	. . . . . 6 ⊢ ((¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐵 ≠ 𝐷) ↔ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))
18	14, 17	bitr2i 276	. . . . 5 ⊢ ((𝐵 = 𝐶 ∨ 𝐵 = 𝐷) ↔ ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷))
19	13, 18	anbi12i 628	. . . 4 ⊢ (((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)) ↔ (¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)))
20	8, 19	imbitrdi 251	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → (¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷))))
21		pm4.56 990	. . 3 ⊢ ((¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) ↔ ¬ ((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)))
22	20, 21	imbitrdi 251	. 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ¬ ((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷))))
23	22	necon2ad 2940	1 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → (((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  {cpr 4579

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-v 3438  df-un 3908  df-sn 4578  df-pr 4580

This theorem is referenced by:  prnebg  4807  opthhausdorff  5460  symg2bas  19272  m2detleib  22516  umgrvad2edg  29162  usgrexmpldifpr  29207  usgrexmpl1lem  48025  usgrexmpl2lem  48030  usgrexmpl2nb0  48035  usgrexmpl2nb1  48036  usgrexmpl2nb2  48037  usgrexmpl2nb3  48038  usgrexmpl2nb4  48039  usgrexmpl2nb5  48040  gpg5nbgrvtx03starlem1  48072  gpg5nbgrvtx03starlem2  48073  gpg5nbgrvtx03starlem3  48074  gpg5nbgrvtx13starlem1  48075  gpg5nbgrvtx13starlem2  48076  gpg5nbgrvtx13starlem3  48077  gpgprismgr4cycllem2  48100  gpg5edgnedg  48134  zlmodzxzldeplem  48503  line2x  48759  inlinecirc02plem  48791