ordssun - Metamath Proof Explorer

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Theorem ordssun 6488

Description: Property of a subclass of the maximum (i.e. union) of two ordinals. (Contributed by NM, 28-Nov-2003.)

**Assertion**
Ref	Expression
ordssun	⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))

**Proof of Theorem ordssun**
Step	Hyp	Ref	Expression
1		ordtri2or2 6485	. . 3 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵))
2		ssequn1 4187	. . . . . 6 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∪ 𝐶) = 𝐶)
3		sseq2 3939	. . . . . 6 ⊢ ((𝐵 ∪ 𝐶) = 𝐶 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐶))
4	2, 3	sylbi 220	. . . . 5 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐶))
5		olc 882	. . . . 5 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶))
6	4, 5	biimtrdi 256	. . . 4 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ⊆ (𝐵 ∪ 𝐶) → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))
7		ssequn2 4190	. . . . . 6 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐶) = 𝐵)
8		sseq2 3939	. . . . . 6 ⊢ ((𝐵 ∪ 𝐶) = 𝐵 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐵))
9	7, 8	sylbi 220	. . . . 5 ⊢ (𝐶 ⊆ 𝐵 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐵))
10		orc 881	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶))
11	9, 10	biimtrdi 256	. . . 4 ⊢ (𝐶 ⊆ 𝐵 → (𝐴 ⊆ (𝐵 ∪ 𝐶) → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))
12	6, 11	jaoi 871	. . 3 ⊢ ((𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵) → (𝐴 ⊆ (𝐵 ∪ 𝐶) → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))
13	1, 12	syl 18	. 2 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ⊆ (𝐵 ∪ 𝐶) → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))
14		ssun 4196	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶))
15	13, 14	impbid1 228	1 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ⊆ wss 3900 ∪ cun 4152 Ord word 6382

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-ext 2735 ax-sep 5253 ax-pr 5396

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-sb 2102 df-clab 2742 df-cleq 2755 df-clel 2839 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 5172 df-tr 5217 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-we 5611 df-ord 6386

This theorem is used by: ordsucun 7863