Theorem prnebg 4817

Description: A (proper) pair is not equal to another (maybe improper) pair if and only if an element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 16-Jan-2018.)

Assertion

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

Proof of Theorem prnebg

Step	Hyp	Ref	Expression
1		prneimg 4815	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → (((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}))
2	1	3adant3 1148	. 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → (((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}))
3		ioran 999	. . . . 5 ⊢ (¬ ((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) ↔ (¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)))
4		ianor 997	. . . . . . 7 ⊢ (¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ↔ (¬ 𝐴 ≠ 𝐶 ∨ ¬ 𝐴 ≠ 𝐷))
5		nne 2964	. . . . . . . 8 ⊢ (¬ 𝐴 ≠ 𝐶 ↔ 𝐴 = 𝐶)
6		nne 2964	. . . . . . . 8 ⊢ (¬ 𝐴 ≠ 𝐷 ↔ 𝐴 = 𝐷)
7	5, 6	orbi12i 927	. . . . . . 7 ⊢ ((¬ 𝐴 ≠ 𝐶 ∨ ¬ 𝐴 ≠ 𝐷) ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
8	4, 7	bitri 278	. . . . . 6 ⊢ (¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
9		ianor 997	. . . . . . 7 ⊢ (¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ↔ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐵 ≠ 𝐷))
10		nne 2964	. . . . . . . 8 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
11		nne 2964	. . . . . . . 8 ⊢ (¬ 𝐵 ≠ 𝐷 ↔ 𝐵 = 𝐷)
12	10, 11	orbi12i 927	. . . . . . 7 ⊢ ((¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐵 ≠ 𝐷) ↔ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))
13	9, 12	bitri 278	. . . . . 6 ⊢ (¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ↔ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))
14	8, 13	anbi12i 639	. . . . 5 ⊢ ((¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) ↔ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)))
15	3, 14	bitri 278	. . . 4 ⊢ (¬ ((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) ↔ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)))
16		anddi 1026	. . . . 5 ⊢ (((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)) ↔ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) ∨ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐷))))
17		eqtr3 2787	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵)
18		eqneqall 2971	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
19	17, 18	syl 18	. . . . . . . . 9 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
20		preq12 4697	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
21	20	a1d 26	. . . . . . . . 9 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
22	19, 21	jaoi 870	. . . . . . . 8 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
23		preq12 4697	. . . . . . . . . . 11 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐷, 𝐶})
24		prcom 4694	. . . . . . . . . . 11 ⊢ {𝐷, 𝐶} = {𝐶, 𝐷}
25	23, 24	eqtrdi 2816	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
26	25	a1d 26	. . . . . . . . 9 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
27		eqtr3 2787	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐷) → 𝐴 = 𝐵)
28	27, 18	syl 18	. . . . . . . . 9 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐷) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
29	26, 28	jaoi 870	. . . . . . . 8 ⊢ (((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐷)) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
30	22, 29	jaoi 870	. . . . . . 7 ⊢ ((((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) ∨ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐷))) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
31	30	com12 33	. . . . . 6 ⊢ (𝐴 ≠ 𝐵 → ((((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) ∨ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐷))) → {𝐴, 𝐵} = {𝐶, 𝐷}))
32	31	3ad2ant3 1151	. . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → ((((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) ∨ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐷))) → {𝐴, 𝐵} = {𝐶, 𝐷}))
33	16, 32	biimtrid 245	. . . 4 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → (((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)) → {𝐴, 𝐵} = {𝐶, 𝐷}))
34	15, 33	biimtrid 245	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → (¬ ((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) → {𝐴, 𝐵} = {𝐶, 𝐷}))
35	34	necon1ad 2977	. 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} → ((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷))))
36	2, 35	impbid 215	1 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → (((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) ↔ {𝐴, 𝐵} ≠ {𝐶, 𝐷}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  {cpr 4587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-un 3912  df-sn 4586  df-pr 4588

This theorem is referenced by:  zlmodzxznm  49128