Theorem poprelb 44928

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 1135	. . 3 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
2		an3 655	. . . . 5 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → (Rel 𝑅 ∧ 𝐶𝑅𝐷))
3	2	3adant2 1129	. . . 4 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → (Rel 𝑅 ∧ 𝐶𝑅𝐷))
4		brrelex12 5638	. . . 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 4789	. . 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 583	. 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 5083	. . . . . . . . . . 11 ⊢ ((𝐵 = 𝐶 ∧ 𝐴 = 𝐷) → (𝐵𝑅𝐴 ↔ 𝐶𝑅𝐷))
10	9	ancoms 458	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐵𝑅𝐴 ↔ 𝐶𝑅𝐷))
11	10	bicomd 222	. . . . . . . . 9 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐶𝑅𝐷 ↔ 𝐵𝑅𝐴))
12	11	anbi2d 628	. . . . . . . 8 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → ((𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)))
13		po2nr 5516	. . . . . . . . . . . 12 ⊢ ((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴))
14	13	adantll 710	. . . . . . . . . . 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 412	. . . . . . . . 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	syl6bi 252	. . . . . . 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 1109	. . . 4 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
22	8, 21	jaod 855	. . 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 863	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))
24	22, 23	impbid1 224	. 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 278	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 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109  Vcvv 3430  {cpr 4568   class class class wbr 5078   Po wpo 5500  Rel wrel 5593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-po 5502  df-xp 5594  df-rel 5595

This theorem is referenced by: (None)