Theorem poprelb 48306

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 1155	. . 3 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
2		an3 672	. . . . 5 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → (Rel 𝑅 ∧ 𝐶𝑅𝐷))
3	2	3adant2 1149	. . . 4 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → (Rel 𝑅 ∧ 𝐶𝑅𝐷))
4		brrelex12 5718	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝐶𝑅𝐷) → (𝐶 ∈ V ∧ 𝐷 ∈ V))
5	3, 4	syl 18	. . 3 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → (𝐶 ∈ V ∧ 𝐷 ∈ V))
6		preq12bg 4823	. . 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 596	. 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 25	. . . 4 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
9		breq12 5119	. . . . . . . . . . 11 ⊢ ((𝐵 = 𝐶 ∧ 𝐴 = 𝐷) → (𝐵𝑅𝐴 ↔ 𝐶𝑅𝐷))
10	9	ancoms 464	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐵𝑅𝐴 ↔ 𝐶𝑅𝐷))
11	10	bicomd 226	. . . . . . . . 9 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐶𝑅𝐷 ↔ 𝐵𝑅𝐴))
12	11	anbi2d 642	. . . . . . . 8 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → ((𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)))
13		po2nr 5588	. . . . . . . . . . . 12 ⊢ ((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴))
14	13	adantll 727	. . . . . . . . . . 11 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴))
15	14	pm2.21d 122	. . . . . . . . . 10 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
16	15	ex 418	. . . . . . . . 9 ⊢ ((Rel 𝑅 ∧ 𝑅 Po 𝑋) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))))
17	16	com13 89	. . . . . . . 8 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((Rel 𝑅 ∧ 𝑅 Po 𝑋) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))))
18	12, 17	biimtrdi 256	. . . . . . 7 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → ((𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((Rel 𝑅 ∧ 𝑅 Po 𝑋) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))))
19	18	com23 87	. . . . . 6 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷) → ((Rel 𝑅 ∧ 𝑅 Po 𝑋) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))))
20	19	com14 97	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝑅 Po 𝑋) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷) → ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))))
21	20	3imp 1128	. . . 4 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
22	8, 21	jaod 873	. . 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 881	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))
24	22, 23	impbid1 228	. 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 282	1 ⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  Vcvv 3458  {cpr 4596   class class class wbr 5114   Po wpo 5572  Rel wrel 5671

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-po 5574  df-xp 5672  df-rel 5673

This theorem is used by: (None)