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Theorem ordssun 6421

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 6418	. . 3 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵))
2		ssequn1 4138	. . . . . 6 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∪ 𝐶) = 𝐶)
3		sseq2 3960	. . . . . 6 ⊢ ((𝐵 ∪ 𝐶) = 𝐶 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐶))
4	2, 3	sylbi 217	. . . . 5 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐶))
5		olc 868	. . . . 5 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶))
6	4, 5	biimtrdi 253	. . . 4 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ⊆ (𝐵 ∪ 𝐶) → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))
7		ssequn2 4141	. . . . . 6 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐶) = 𝐵)
8		sseq2 3960	. . . . . 6 ⊢ ((𝐵 ∪ 𝐶) = 𝐵 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐵))
9	7, 8	sylbi 217	. . . . 5 ⊢ (𝐶 ⊆ 𝐵 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐵))
10		orc 867	. . . . 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 857	. . 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 4147	. 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 847 = wceq 1541 ∪ cun 3899 ⊆ wss 3901 Ord word 6316

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-sb 2068 df-clab 2715 df-cleq 2728 df-clel 2811 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-tr 5206 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-ord 6320

This theorem is referenced by: ordsucun 7767