Theorem prnebg 4820

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 4818	. . 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 2960	. . . . . . . 8 ⊢ (¬ 𝐴 ≠ 𝐶 ↔ 𝐴 = 𝐶)
6		nne 2960	. . . . . . . 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 2960	. . . . . . . 8 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
11		nne 2960	. . . . . . . 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 2783	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵)
18		eqneqall 2967	. . . . . . . . . 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 4700	. . . . . . . . . 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 4700	. . . . . . . . . . 11 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐷, 𝐶})
24		prcom 4697	. . . . . . . . . . 11 ⊢ {𝐷, 𝐶} = {𝐶, 𝐷}
25	23, 24	eqtrdi 2812	. . . . . . . . . 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 2783	. . . . . . . . . 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 2973	. 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 1568   ∈ wcel 2141   ≠ wne 2956  {cpr 4590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3455  df-un 3909  df-sn 4589  df-pr 4591

This theorem is referenced by:  zlmodzxznm  49244