Theorem preq12nebg 4832

Description: Equality relationship for two proper unordered pairs. (Contributed by AV, 12-Jun-2022.)

Assertion

Ref	Expression
preq12nebg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))

Proof of Theorem preq12nebg

Step	Hyp	Ref	Expression
1		3simpa 1164	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊))
2	1	anim1i 626	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)))
3	2	ancoms 463	. . . 4 ⊢ (((𝐶 ∈ V ∧ 𝐷 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)))
4		preq12bg 4822	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
5	3, 4	syl 18	. . 3 ⊢ (((𝐶 ∈ V ∧ 𝐷 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
6	5	ex 417	. 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))))
7		ianor 997	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V))
8		prneprprc 4830	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
9	8	ancoms 463	. . . . . . 7 ⊢ ((¬ 𝐶 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
10		eqneqall 2975	. . . . . . 7 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
11	9, 10	syl5com 32	. . . . . 6 ⊢ ((¬ 𝐶 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
12		prneprprc 4830	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ ¬ 𝐷 ∈ V) → {𝐴, 𝐵} ≠ {𝐷, 𝐶})
13	12	ancoms 463	. . . . . . 7 ⊢ ((¬ 𝐷 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → {𝐴, 𝐵} ≠ {𝐷, 𝐶})
14		prcom 4703	. . . . . . . . 9 ⊢ {𝐶, 𝐷} = {𝐷, 𝐶}
15	14	eqeq2i 2782	. . . . . . . 8 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐷, 𝐶})
16		eqneqall 2975	. . . . . . . 8 ⊢ ({𝐴, 𝐵} = {𝐷, 𝐶} → ({𝐴, 𝐵} ≠ {𝐷, 𝐶} → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
17	15, 16	sylbi 220	. . . . . . 7 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → ({𝐴, 𝐵} ≠ {𝐷, 𝐶} → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
18	13, 17	syl5com 32	. . . . . 6 ⊢ ((¬ 𝐷 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
19	11, 18	jaoian 971	. . . . 5 ⊢ (((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
20		preq12 4706	. . . . . 6 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
21		preq12 4706	. . . . . . 7 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐷, 𝐶})
22		prcom 4703	. . . . . . 7 ⊢ {𝐷, 𝐶} = {𝐶, 𝐷}
23	21, 22	eqtrdi 2820	. . . . . 6 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
24	20, 23	jaoi 870	. . . . 5 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) → {𝐴, 𝐵} = {𝐶, 𝐷})
25	19, 24	impbid1 228	. . . 4 ⊢ (((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
26	25	ex 417	. . 3 ⊢ ((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))))
27	7, 26	sylbi 220	. 2 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))))
28	6, 27	pm2.61i 184	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  Vcvv 3463  {cpr 4596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-v 3465  df-dif 3916  df-un 3918  df-nul 4295  df-sn 4595  df-pr 4597

This theorem is referenced by:  prel12g  4833  opthhausdorff  5501  preleqg  9584  pr2cv  44200