Theorem poprelb 48130

Description: Equality for unordered pairs with partially ordered elements. (Contributed by AV, 9-Jul-2023.)

Assertion

Ref	Expression
poprelb	⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Proof of Theorem poprelb

Step	Hyp	Ref	Expression
1		simp2 1150	. . 3 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
2		an3 669	. . . . 5 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → (Rel 𝑅 ∧ 𝐶𝑅𝐷))
3	2	3adant2 1144	. . . 4 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → (Rel 𝑅 ∧ 𝐶𝑅𝐷))
4		brrelex12 5699	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝐶𝑅𝐷) → (𝐶 ∈ V ∧ 𝐷 ∈ V))
5	3, 4	syl 17	. . 3 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → (𝐶 ∈ V ∧ 𝐷 ∈ V))
6		preq12bg 4811	. . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
7	1, 5, 6	syl2anc 593	. 2 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
8		idd 24	. . . 4 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
9		breq12 5105	. . . . . . . . . . 11 ⊢ ((𝐵 = 𝐶 ∧ 𝐴 = 𝐷) → (𝐵𝑅𝐴 ↔ 𝐶𝑅𝐷))
10	9	ancoms 462	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐵𝑅𝐴 ↔ 𝐶𝑅𝐷))
11	10	bicomd 225	. . . . . . . . 9 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐶𝑅𝐷 ↔ 𝐵𝑅𝐴))
12	11	anbi2d 639	. . . . . . . 8 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → ((𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)))
13		po2nr 5569	. . . . . . . . . . . 12 ⊢ ((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴))
14	13	adantll 724	. . . . . . . . . . 11 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴))
15	14	pm2.21d 121	. . . . . . . . . 10 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
16	15	ex 416	. . . . . . . . 9 ⊢ ((Rel 𝑅 ∧ 𝑅 Po 𝑋) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))))
17	16	com13 88	. . . . . . . 8 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((Rel 𝑅 ∧ 𝑅 Po 𝑋) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))))
18	12, 17	biimtrdi 255	. . . . . . 7 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → ((𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((Rel 𝑅 ∧ 𝑅 Po 𝑋) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))))
19	18	com23 86	. . . . . 6 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷) → ((Rel 𝑅 ∧ 𝑅 Po 𝑋) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))))
20	19	com14 96	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝑅 Po 𝑋) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷) → ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))))
21	20	3imp 1123	. . . 4 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
22	8, 21	jaod 870	. . 3 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → (((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
23		orc 878	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))
24	22, 23	impbid1 227	. 2 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → (((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
25	7, 24	bitrd 281	1 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  Vcvv 3454  {cpr 4584   class class class wbr 5100   Po wpo 5553  Rel wrel 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-po 5555  df-xp 5653  df-rel 5654

This theorem is referenced by: (None)