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Theorem ordssun 6497

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 6494	. . 3 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵))
2		ssequn1 4209	. . . . . 6 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∪ 𝐶) = 𝐶)
3		sseq2 4035	. . . . . 6 ⊢ ((𝐵 ∪ 𝐶) = 𝐶 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐶))
4	2, 3	sylbi 217	. . . . 5 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐶))
5		olc 867	. . . . 5 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶))
6	4, 5	biimtrdi 253	. . . 4 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ⊆ (𝐵 ∪ 𝐶) → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))
7		ssequn2 4212	. . . . . 6 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐶) = 𝐵)
8		sseq2 4035	. . . . . 6 ⊢ ((𝐵 ∪ 𝐶) = 𝐵 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐵))
9	7, 8	sylbi 217	. . . . 5 ⊢ (𝐶 ⊆ 𝐵 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐵))
10		orc 866	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶))
11	9, 10	biimtrdi 253	. . . 4 ⊢ (𝐶 ⊆ 𝐵 → (𝐴 ⊆ (𝐵 ∪ 𝐶) → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))
12	6, 11	jaoi 856	. . 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 4218	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶))
15	13, 14	impbid1 225	1 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 846 = wceq 1537 ∪ cun 3974 ⊆ wss 3976 Ord word 6394

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-tr 5284 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-ord 6398

This theorem is referenced by: ordsucun 7861