Theorem onunel 6468

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 4179	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
2	1	biimpi 215	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) = 𝐵)
3	2	eleq1d 2816	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
4	3	adantl 480	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
5		ontr2 6410	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
6	5	3adant2 1129	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
7	6	expdimp 451	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐵 ∈ 𝐶 → 𝐴 ∈ 𝐶))
8	7	pm4.71rd 561	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐵 ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
9	4, 8	bitrd 278	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
10		ssequn2 4182	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
11	10	biimpi 215	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴)
12	11	eleq1d 2816	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
13	12	adantl 480	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 ⊆ 𝐴) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
14		ontr2 6410	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐵 ∈ 𝐶))
15	14	3adant1 1128	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐵 ∈ 𝐶))
16	15	expdimp 451	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∈ 𝐶 → 𝐵 ∈ 𝐶))
17	16	pm4.71d 560	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
18	13, 17	bitrd 278	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 ⊆ 𝐴) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
19		eloni 6373	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
20		eloni 6373	. . . 4 ⊢ (𝐵 ∈ On → Ord 𝐵)
21		ordtri2or2 6462	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
22	19, 20, 21	syl2an 594	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
23	22	3adant3 1130	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
24	9, 18, 23	mpjaodan 955	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   ∪ cun 3945   ⊆ wss 3947  Ord word 6362  Oncon0 6363

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6366  df-on 6367

This theorem is referenced by:  addsproplem2  27692  negsproplem2  27742  mulsproplem5  27815  mulsproplem6  27816  mulsproplem7  27817  mulsproplem8  27818