Theorem prnebg 4857

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 4856	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → (((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}))
2	1	3adant3 1133	. 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 983	. . . . 5 ⊢ (¬ ((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) ↔ (¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)))
4		ianor 981	. . . . . . 7 ⊢ (¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ↔ (¬ 𝐴 ≠ 𝐶 ∨ ¬ 𝐴 ≠ 𝐷))
5		nne 2945	. . . . . . . 8 ⊢ (¬ 𝐴 ≠ 𝐶 ↔ 𝐴 = 𝐶)
6		nne 2945	. . . . . . . 8 ⊢ (¬ 𝐴 ≠ 𝐷 ↔ 𝐴 = 𝐷)
7	5, 6	orbi12i 914	. . . . . . 7 ⊢ ((¬ 𝐴 ≠ 𝐶 ∨ ¬ 𝐴 ≠ 𝐷) ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
8	4, 7	bitri 275	. . . . . 6 ⊢ (¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
9		ianor 981	. . . . . . 7 ⊢ (¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ↔ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐵 ≠ 𝐷))
10		nne 2945	. . . . . . . 8 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
11		nne 2945	. . . . . . . 8 ⊢ (¬ 𝐵 ≠ 𝐷 ↔ 𝐵 = 𝐷)
12	10, 11	orbi12i 914	. . . . . . 7 ⊢ ((¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐵 ≠ 𝐷) ↔ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))
13	9, 12	bitri 275	. . . . . 6 ⊢ (¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ↔ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))
14	8, 13	anbi12i 628	. . . . 5 ⊢ ((¬ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ ¬ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) ↔ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)))
15	3, 14	bitri 275	. . . 4 ⊢ (¬ ((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) ↔ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)))
16		anddi 1010	. . . . 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 2759	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵)
18		eqneqall 2952	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
19	17, 18	syl 17	. . . . . . . . 9 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
20		preq12 4740	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
21	20	a1d 25	. . . . . . . . 9 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
22	19, 21	jaoi 856	. . . . . . . 8 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
23		preq12 4740	. . . . . . . . . . 11 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐷, 𝐶})
24		prcom 4737	. . . . . . . . . . 11 ⊢ {𝐷, 𝐶} = {𝐶, 𝐷}
25	23, 24	eqtrdi 2789	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
26	25	a1d 25	. . . . . . . . 9 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
27		eqtr3 2759	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐷) → 𝐴 = 𝐵)
28	27, 18	syl 17	. . . . . . . . 9 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐷) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
29	26, 28	jaoi 856	. . . . . . . 8 ⊢ (((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐷)) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
30	22, 29	jaoi 856	. . . . . . 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 32	. . . . . 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 1136	. . . . 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 241	. . . 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 241	. . 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 2958	. 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 211	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 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  {cpr 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-v 3477  df-un 3954  df-sn 4630  df-pr 4632

This theorem is referenced by:  zlmodzxznm  47178