Theorem brr0d 36001

Description: Condition for two ordered pairs to be related by 𝑅₀. (Contributed by BTernaryTau, 19-Sep-2026.)

Hypotheses

Ref	Expression
brr0d.1	⊢ (𝜑 → 𝐴 ∈ On)
brr0d.2	⊢ (𝜑 → 𝐵 ∈ On)
brr0d.3	⊢ (𝜑 → 𝐶 ∈ On)
brr0d.4	⊢ (𝜑 → 𝐷 ∈ On)

Assertion

Ref	Expression
brr0d	⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝑅0⟨𝐶, 𝐷⟩ ↔ ((𝐴 ∪ 𝐵) ∈ (𝐶 ∪ 𝐷) ∨ ((𝐴 ∪ 𝐵) = (𝐶 ∪ 𝐷) ∧ ⟨𝐴, 𝐵⟩LexOrd⟨𝐶, 𝐷⟩))))

Proof of Theorem brr0d

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brr0d.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ On)
2		brr0d.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ On)
3	1, 2	opelxpd 5694	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (On × On))
4		brr0d.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ On)
5		brr0d.4	. . . 4 ⊢ (𝜑 → 𝐷 ∈ On)
6	4, 5	opelxpd 5694	. . 3 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (On × On))
7	3, 6	jca 521	. 2 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ (On × On) ∧ ⟨𝐶, 𝐷⟩ ∈ (On × On)))
8		df-r0 35990	. . . 4 ⊢ 𝑅0 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ (((1st ‘𝑥) ∪ (2nd ‘𝑥)) ∈ ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ∨ (((1st ‘𝑥) ∪ (2nd ‘𝑥)) = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ∧ 𝑥LexOrd𝑦)))}
9	8	a1i 11	. . 3 ⊢ (𝜑 → 𝑅0 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ (((1st ‘𝑥) ∪ (2nd ‘𝑥)) ∈ ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ∨ (((1st ‘𝑥) ∪ (2nd ‘𝑥)) = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ∧ 𝑥LexOrd𝑦)))})
10		fveq2 6908	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
11		op1stg 8037	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
12	1, 2, 11	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
13	10, 12	sylan9eqr 2820	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) → (1st ‘𝑥) = 𝐴)
14		fveq2 6908	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
15		op2ndg 8038	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
16	1, 2, 15	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
17	14, 16	sylan9eqr 2820	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) → (2nd ‘𝑥) = 𝐵)
18	13, 17	uneq12d 4171	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) → ((1st ‘𝑥) ∪ (2nd ‘𝑥)) = (𝐴 ∪ 𝐵))
19	18	adantrr 730	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = ⟨𝐶, 𝐷⟩)) → ((1st ‘𝑥) ∪ (2nd ‘𝑥)) = (𝐴 ∪ 𝐵))
20		fveq2 6908	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (1st ‘𝑦) = (1st ‘⟨𝐶, 𝐷⟩))
21		op1stg 8037	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
22	4, 5, 21	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
23	20, 22	sylan9eqr 2820	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = ⟨𝐶, 𝐷⟩) → (1st ‘𝑦) = 𝐶)
24		fveq2 6908	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (2nd ‘𝑦) = (2nd ‘⟨𝐶, 𝐷⟩))
25		op2ndg 8038	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
26	4, 5, 25	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
27	24, 26	sylan9eqr 2820	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = ⟨𝐶, 𝐷⟩) → (2nd ‘𝑦) = 𝐷)
28	23, 27	uneq12d 4171	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = ⟨𝐶, 𝐷⟩) → ((1st ‘𝑦) ∪ (2nd ‘𝑦)) = (𝐶 ∪ 𝐷))
29	28	adantrl 729	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = ⟨𝐶, 𝐷⟩)) → ((1st ‘𝑦) ∪ (2nd ‘𝑦)) = (𝐶 ∪ 𝐷))
30	19, 29	eleq12d 2858	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = ⟨𝐶, 𝐷⟩)) → (((1st ‘𝑥) ∪ (2nd ‘𝑥)) ∈ ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↔ (𝐴 ∪ 𝐵) ∈ (𝐶 ∪ 𝐷)))
31	19, 29	eqeq12d 2779	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = ⟨𝐶, 𝐷⟩)) → (((1st ‘𝑥) ∪ (2nd ‘𝑥)) = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↔ (𝐴 ∪ 𝐵) = (𝐶 ∪ 𝐷)))
32		breq12 5111	. . . . . 6 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = ⟨𝐶, 𝐷⟩) → (𝑥LexOrd𝑦 ↔ ⟨𝐴, 𝐵⟩LexOrd⟨𝐶, 𝐷⟩))
33	32	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = ⟨𝐶, 𝐷⟩)) → (𝑥LexOrd𝑦 ↔ ⟨𝐴, 𝐵⟩LexOrd⟨𝐶, 𝐷⟩))
34	31, 33	anbi12d 644	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = ⟨𝐶, 𝐷⟩)) → ((((1st ‘𝑥) ∪ (2nd ‘𝑥)) = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ∧ 𝑥LexOrd𝑦) ↔ ((𝐴 ∪ 𝐵) = (𝐶 ∪ 𝐷) ∧ ⟨𝐴, 𝐵⟩LexOrd⟨𝐶, 𝐷⟩)))
35	30, 34	orbi12d 932	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = ⟨𝐶, 𝐷⟩)) → ((((1st ‘𝑥) ∪ (2nd ‘𝑥)) ∈ ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ∨ (((1st ‘𝑥) ∪ (2nd ‘𝑥)) = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ∧ 𝑥LexOrd𝑦)) ↔ ((𝐴 ∪ 𝐵) ∈ (𝐶 ∪ 𝐷) ∨ ((𝐴 ∪ 𝐵) = (𝐶 ∪ 𝐷) ∧ ⟨𝐴, 𝐵⟩LexOrd⟨𝐶, 𝐷⟩))))
36	9, 35	brab2d 5516	. 2 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝑅0⟨𝐶, 𝐷⟩ ↔ ((⟨𝐴, 𝐵⟩ ∈ (On × On) ∧ ⟨𝐶, 𝐷⟩ ∈ (On × On)) ∧ ((𝐴 ∪ 𝐵) ∈ (𝐶 ∪ 𝐷) ∨ ((𝐴 ∪ 𝐵) = (𝐶 ∪ 𝐷) ∧ ⟨𝐴, 𝐵⟩LexOrd⟨𝐶, 𝐷⟩)))))
37	7, 36	mpbirand 720	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝑅0⟨𝐶, 𝐷⟩ ↔ ((𝐴 ∪ 𝐵) ∈ (𝐶 ∪ 𝐷) ∨ ((𝐴 ∪ 𝐵) = (𝐶 ∪ 𝐷) ∧ ⟨𝐴, 𝐵⟩LexOrd⟨𝐶, 𝐷⟩))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ∪ cun 4152  ⟨cop 4838   class class class wbr 5106  {copab 5170   × cxp 5653  Oncon0 6380  ‘cfv 6558  1^st c1st 8023  2^nd c2nd 8024  LexOrdclexo 35980  𝑅₀cr0 35981

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

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-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fv 6566  df-1st 8025  df-2nd 8026  df-r0 35990

This theorem is used by:  r0maxss  36003  j00smo  36005  maxssj0  36006