Theorem onunel 6442

Description: The union of two ordinals is in a third iff both of the first two are. (Contributed by Scott Fenton, 10-Sep-2024.)

Assertion

Ref	Expression
onunel	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))

Proof of Theorem onunel

Step	Hyp	Ref	Expression
1		ssequn1 4152	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
2	1	biimpi 216	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) = 𝐵)
3	2	eleq1d 2814	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
4	3	adantl 481	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
5		ontr2 6383	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
6	5	3adant2 1131	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
7	6	expdimp 452	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐵 ∈ 𝐶 → 𝐴 ∈ 𝐶))
8	7	pm4.71rd 562	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐵 ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
9	4, 8	bitrd 279	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
10		ssequn2 4155	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
11	10	biimpi 216	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴)
12	11	eleq1d 2814	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
13	12	adantl 481	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 ⊆ 𝐴) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
14		ontr2 6383	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐵 ∈ 𝐶))
15	14	3adant1 1130	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐵 ∈ 𝐶))
16	15	expdimp 452	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∈ 𝐶 → 𝐵 ∈ 𝐶))
17	16	pm4.71d 561	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
18	13, 17	bitrd 279	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 ⊆ 𝐴) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
19		eloni 6345	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
20		eloni 6345	. . . 4 ⊢ (𝐵 ∈ On → Ord 𝐵)
21		ordtri2or2 6436	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
22	19, 20, 21	syl2an 596	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
23	22	3adant3 1132	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
24	9, 18, 23	mpjaodan 960	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∪ cun 3915   ⊆ wss 3917  Ord word 6334  Oncon0 6335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-tr 5218  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-ord 6338  df-on 6339

This theorem is referenced by:  addsproplem2  27884  negsproplem2  27942  mulsproplem5  28030  mulsproplem6  28031  mulsproplem7  28032  mulsproplem8  28033