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Theorem ordssun 6476

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 6473	. . 3 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵))
2		ssequn1 4182	. . . . . 6 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∪ 𝐶) = 𝐶)
3		sseq2 4008	. . . . . 6 ⊢ ((𝐵 ∪ 𝐶) = 𝐶 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐶))
4	2, 3	sylbi 216	. . . . 5 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐶))
5		olc 866	. . . . 5 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶))
6	4, 5	biimtrdi 252	. . . 4 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ⊆ (𝐵 ∪ 𝐶) → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))
7		ssequn2 4185	. . . . . 6 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐶) = 𝐵)
8		sseq2 4008	. . . . . 6 ⊢ ((𝐵 ∪ 𝐶) = 𝐵 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐵))
9	7, 8	sylbi 216	. . . . 5 ⊢ (𝐶 ⊆ 𝐵 → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐵))
10		orc 865	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶))
11	9, 10	biimtrdi 252	. . . 4 ⊢ (𝐶 ⊆ 𝐵 → (𝐴 ⊆ (𝐵 ∪ 𝐶) → (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))
12	6, 11	jaoi 855	. . 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 4191	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶))
15	13, 14	impbid1 224	1 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∪ cun 3947 ⊆ wss 3949 Ord word 6373

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-tr 5270 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-ord 6377

This theorem is referenced by: ordsucun 7834