Theorem prneimg 4604

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 4602	. . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
2		orddi 1039	. . . . . 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 785	. . . . . . 7 ⊢ ((((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐴 = 𝐶 ∨ 𝐵 = 𝐶)) ∧ ((𝐵 = 𝐷 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐷 ∨ 𝐵 = 𝐶))) → (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
4		pm1.4 902	. . . . . . . 8 ⊢ ((𝐵 = 𝐷 ∨ 𝐵 = 𝐶) → (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))
5	4	ad2antll 722	. . . . . . 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 509	. . . . . 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 209	. . . . 5 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) → ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)))
8	1, 7	syl6bi 245	. . . 4 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))))
9		ianor 1011	. . . . . 6 ⊢ (¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ↔ (¬ 𝐴 ≠ 𝐶 ∨ ¬ 𝐴 ≠ 𝐷))
10		nne 3004	. . . . . . 7 ⊢ (¬ 𝐴 ≠ 𝐶 ↔ 𝐴 = 𝐶)
11		nne 3004	. . . . . . 7 ⊢ (¬ 𝐴 ≠ 𝐷 ↔ 𝐴 = 𝐷)
12	10, 11	orbi12i 945	. . . . . 6 ⊢ ((¬ 𝐴 ≠ 𝐶 ∨ ¬ 𝐴 ≠ 𝐷) ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
13	9, 12	bitr2i 268	. . . . 5 ⊢ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ↔ ¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷))
14		ianor 1011	. . . . . 6 ⊢ (¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ↔ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐵 ≠ 𝐷))
15		nne 3004	. . . . . . 7 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
16		nne 3004	. . . . . . 7 ⊢ (¬ 𝐵 ≠ 𝐷 ↔ 𝐵 = 𝐷)
17	15, 16	orbi12i 945	. . . . . 6 ⊢ ((¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐵 ≠ 𝐷) ↔ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))
18	14, 17	bitr2i 268	. . . . 5 ⊢ ((𝐵 = 𝐶 ∨ 𝐵 = 𝐷) ↔ ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷))
19	13, 18	anbi12i 622	. . . 4 ⊢ (((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)) ↔ (¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)))
20	8, 19	syl6ib 243	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → (¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷))))
21		pm4.56 1018	. . 3 ⊢ ((¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) ↔ ¬ ((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)))
22	20, 21	syl6ib 243	. 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ¬ ((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷))))
23	22	necon2ad 3015	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 386   ∨ wo 880   = wceq 1658   ∈ wcel 2166   ≠ wne 3000  {cpr 4400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2804

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-v 3417  df-un 3804  df-sn 4399  df-pr 4401

This theorem is referenced by:  prnebg  4605  opthhausdorff  5204  symg2bas  18169  m2detleib  20806  umgrvad2edg  26510  usgrexmpldifpr  26556  zlmodzxzldeplem  43135  line2x  43307