Theorem r0maxss 35935

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 35758.) (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 35933	. . . 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 6473	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)
12	2, 3, 11	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ On)
13		onun2 6473	. . . 4 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 ∪ 𝐷) ∈ On)
14	4, 5, 13	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐶 ∪ 𝐷) ∈ On)
15		onsseleq 6404	. . 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 1569   ∈ wcel 2144   ⊆ wss 3895   ∪ cun 4149  ⟨cop 4589   class class class wbr 5102  Oncon0 6362  LexOrdclexo 35912  𝑅₀cr0 35913

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-ord 6365  df-on 6366  df-iota 6494  df-fun 6540  df-fv 6546  df-1st 8001  df-2nd 8002  df-r0 35922

This theorem is used by:  j0ltalephlem  35939