Theorem onunel 33359

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 4080	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
2	1	biimpi 219	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) = 𝐵)
3	2	eleq1d 2815	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
4	3	adantl 485	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
5		ontr2 6238	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
6	5	3adant2 1133	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
7	6	expdimp 456	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐵 ∈ 𝐶 → 𝐴 ∈ 𝐶))
8	7	pm4.71rd 566	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐵 ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
9	4, 8	bitrd 282	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
10		ssequn2 4083	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
11	10	biimpi 219	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴)
12	11	eleq1d 2815	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
13	12	adantl 485	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 ⊆ 𝐴) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
14		ontr2 6238	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐵 ∈ 𝐶))
15	14	3adant1 1132	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐵 ∈ 𝐶))
16	15	expdimp 456	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∈ 𝐶 → 𝐵 ∈ 𝐶))
17	16	pm4.71d 565	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
18	13, 17	bitrd 282	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 ⊆ 𝐴) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
19		eloni 6201	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
20		eloni 6201	. . . 4 ⊢ (𝐵 ∈ On → Ord 𝐵)
21		ordtri2or2 6287	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
22	19, 20, 21	syl2an 599	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
23	22	3adant3 1134	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
24	9, 18, 23	mpjaodan 959	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112   ∪ cun 3851   ⊆ wss 3853  Ord word 6190  Oncon0 6191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-11 2160  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-tr 5147  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-ord 6194  df-on 6195

This theorem is referenced by: (None)