Theorem onunel 6500

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 4209	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
2	1	biimpi 216	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) = 𝐵)
3	2	eleq1d 2829	. . . 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 6442	. . . . . 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 4212	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
11	10	biimpi 216	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴)
12	11	eleq1d 2829	. . . 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 6442	. . . . . 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 6405	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
20		eloni 6405	. . . 4 ⊢ (𝐵 ∈ On → Ord 𝐵)
21		ordtri2or2 6494	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
22	19, 20, 21	syl2an 595	. . 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 959	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ∪ cun 3974   ⊆ wss 3976  Ord word 6394  Oncon0 6395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399

This theorem is referenced by:  addsproplem2  28021  negsproplem2  28079  mulsproplem5  28164  mulsproplem6  28165  mulsproplem7  28166  mulsproplem8  28167