Theorem r0maxss 36003

Description: If two ordered pairs are related by 𝑅₀, then the maximum of the first pair is less than or equal to the maximum of the second pair. (The union of two ordinals is proven to be their maximum in ordunmax 35792.) (Contributed by BTernaryTau, 22-Sep-2026.)

Hypotheses

Ref	Expression
r0maxss.1	⊢ (𝜑 → 𝐴 ∈ On)
r0maxss.2	⊢ (𝜑 → 𝐵 ∈ On)
r0maxss.3	⊢ (𝜑 → 𝐶 ∈ On)
r0maxss.4	⊢ (𝜑 → 𝐷 ∈ On)
r0maxss.5	⊢ (𝜑 → ⟨𝐴, 𝐵⟩𝑅0⟨𝐶, 𝐷⟩)

Assertion

Ref	Expression
r0maxss	⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ (𝐶 ∪ 𝐷))

Proof of Theorem r0maxss

Step	Hyp	Ref	Expression
1		r0maxss.5	. . . 4 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩𝑅0⟨𝐶, 𝐷⟩)
2		r0maxss.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ On)
3		r0maxss.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ On)
4		r0maxss.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ On)
5		r0maxss.4	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ On)
6	2, 3, 4, 5	brr0d 36001	. . . 4 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝑅0⟨𝐶, 𝐷⟩ ↔ ((𝐴 ∪ 𝐵) ∈ (𝐶 ∪ 𝐷) ∨ ((𝐴 ∪ 𝐵) = (𝐶 ∪ 𝐷) ∧ ⟨𝐴, 𝐵⟩LexOrd⟨𝐶, 𝐷⟩))))
7	1, 6	mpbid 235	. . 3 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) ∈ (𝐶 ∪ 𝐷) ∨ ((𝐴 ∪ 𝐵) = (𝐶 ∪ 𝐷) ∧ ⟨𝐴, 𝐵⟩LexOrd⟨𝐶, 𝐷⟩)))
8		simpl 488	. . . 4 ⊢ (((𝐴 ∪ 𝐵) = (𝐶 ∪ 𝐷) ∧ ⟨𝐴, 𝐵⟩LexOrd⟨𝐶, 𝐷⟩) → (𝐴 ∪ 𝐵) = (𝐶 ∪ 𝐷))
9	8	orim2i 924	. . 3 ⊢ (((𝐴 ∪ 𝐵) ∈ (𝐶 ∪ 𝐷) ∨ ((𝐴 ∪ 𝐵) = (𝐶 ∪ 𝐷) ∧ ⟨𝐴, 𝐵⟩LexOrd⟨𝐶, 𝐷⟩)) → ((𝐴 ∪ 𝐵) ∈ (𝐶 ∪ 𝐷) ∨ (𝐴 ∪ 𝐵) = (𝐶 ∪ 𝐷)))
10	7, 9	syl 18	. 2 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) ∈ (𝐶 ∪ 𝐷) ∨ (𝐴 ∪ 𝐵) = (𝐶 ∪ 𝐷)))
11		onun2 6491	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)
12	2, 3, 11	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ On)
13		onun2 6491	. . . 4 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 ∪ 𝐷) ∈ On)
14	4, 5, 13	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐶 ∪ 𝐷) ∈ On)
15		onsseleq 6422	. . 3 ⊢ (((𝐴 ∪ 𝐵) ∈ On ∧ (𝐶 ∪ 𝐷) ∈ On) → ((𝐴 ∪ 𝐵) ⊆ (𝐶 ∪ 𝐷) ↔ ((𝐴 ∪ 𝐵) ∈ (𝐶 ∪ 𝐷) ∨ (𝐴 ∪ 𝐵) = (𝐶 ∪ 𝐷))))
16	12, 14, 15	syl2anc 596	. 2 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) ⊆ (𝐶 ∪ 𝐷) ↔ ((𝐴 ∪ 𝐵) ∈ (𝐶 ∪ 𝐷) ∨ (𝐴 ∪ 𝐵) = (𝐶 ∪ 𝐷))))
17	10, 16	mpbird 260	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ (𝐶 ∪ 𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∪ cun 4152  ⟨cop 4838   class class class wbr 5106  Oncon0 6380  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-3or 1104  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-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-ord 6383  df-on 6384  df-iota 6512  df-fun 6560  df-fv 6566  df-1st 8025  df-2nd 8026  df-r0 35990

This theorem is used by:  j0ltalephlem  36007