Theorem prneimg 3566

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 3565	. . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
2		orddi 766	. . . . . 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 495	. . . . . . 7 ⊢ ((((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐴 = 𝐶 ∨ 𝐵 = 𝐶)) ∧ ((𝐵 = 𝐷 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐷 ∨ 𝐵 = 𝐶))) → (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
4		pm1.4 678	. . . . . . . 8 ⊢ ((𝐵 = 𝐷 ∨ 𝐵 = 𝐶) → (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))
5	4	ad2antll 474	. . . . . . 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 300	. . . . . 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 119	. . . . 5 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) → ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)))
8	1, 7	syl6bi 161	. . . 4 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))))
9		oranim 840	. . . . . 6 ⊢ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) → ¬ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐴 = 𝐷))
10		df-ne 2246	. . . . . . 7 ⊢ (𝐴 ≠ 𝐶 ↔ ¬ 𝐴 = 𝐶)
11		df-ne 2246	. . . . . . 7 ⊢ (𝐴 ≠ 𝐷 ↔ ¬ 𝐴 = 𝐷)
12	10, 11	anbi12i 447	. . . . . 6 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐴 = 𝐷))
13	9, 12	sylnibr 634	. . . . 5 ⊢ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) → ¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷))
14		oranim 840	. . . . . 6 ⊢ ((𝐵 = 𝐶 ∨ 𝐵 = 𝐷) → ¬ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐵 = 𝐷))
15		df-ne 2246	. . . . . . 7 ⊢ (𝐵 ≠ 𝐶 ↔ ¬ 𝐵 = 𝐶)
16		df-ne 2246	. . . . . . 7 ⊢ (𝐵 ≠ 𝐷 ↔ ¬ 𝐵 = 𝐷)
17	15, 16	anbi12i 447	. . . . . 6 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ↔ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐵 = 𝐷))
18	14, 17	sylnibr 634	. . . . 5 ⊢ ((𝐵 = 𝐶 ∨ 𝐵 = 𝐷) → ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷))
19	13, 18	anim12i 331	. . . 4 ⊢ (((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)) → (¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)))
20	8, 19	syl6 33	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → (¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷))))
21		pm4.56 839	. . 3 ⊢ ((¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) ↔ ¬ ((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)))
22	20, 21	syl6ib 159	. 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ¬ ((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷))))
23	22	necon2ad 2302	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 102   ∨ wo 661   = wceq 1284   ∈ wcel 1433   ≠ wne 2245  {cpr 3399

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405

This theorem is referenced by: (None)