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Theorem ordssun 5744

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 5740	. . 3 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵))
2		ssequn1 3745	. . . . . 6 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∪ 𝐶) = 𝐶)
3		sseq2 3590	. . . . . 6 ⊢ ((𝐵 ∪ 𝐶) = 𝐶 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐶))
4	2, 3	sylbi 206	. . . . 5 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐶))
5		olc 398	. . . . 5 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶))
6	4, 5	syl6bi 242	. . . 4 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ⊆ (𝐵 ∪ 𝐶) → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))
7		ssequn2 3748	. . . . . 6 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐶) = 𝐵)
8		sseq2 3590	. . . . . 6 ⊢ ((𝐵 ∪ 𝐶) = 𝐵 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐵))
9	7, 8	sylbi 206	. . . . 5 ⊢ (𝐶 ⊆ 𝐵 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐵))
10		orc 399	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶))
11	9, 10	syl6bi 242	. . . 4 ⊢ (𝐶 ⊆ 𝐵 → (𝐴 ⊆ (𝐵 ∪ 𝐶) → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))
12	6, 11	jaoi 393	. . 3 ⊢ ((𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵) → (𝐴 ⊆ (𝐵 ∪ 𝐶) → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))
13	1, 12	syl 17	. 2 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ⊆ (𝐵 ∪ 𝐶) → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))
14		ssun 3754	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶))
15	13, 14	impbid1 214	1 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∪ cun 3538 ⊆ wss 3540 Ord word 5639

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643

This theorem is referenced by: ordsucun 6917