Theorem onunel 6469

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 4180	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
2	1	biimpi 215	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) = 𝐵)
3	2	eleq1d 2818	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
4	3	adantl 482	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
5		ontr2 6411	. . . . . 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 453	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐵 ∈ 𝐶 → 𝐴 ∈ 𝐶))
8	7	pm4.71rd 563	. . 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 4183	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
11	10	biimpi 215	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴)
12	11	eleq1d 2818	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
13	12	adantl 482	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 ⊆ 𝐴) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
14		ontr2 6411	. . . . . 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 453	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∈ 𝐶 → 𝐵 ∈ 𝐶))
17	16	pm4.71d 562	. . 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 6374	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
20		eloni 6374	. . . 4 ⊢ (𝐵 ∈ On → Ord 𝐵)
21		ordtri2or2 6463	. . . 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 957	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ∪ cun 3946   ⊆ wss 3948  Ord word 6363  Oncon0 6364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368

This theorem is referenced by:  addsproplem2  27680  negsproplem2  27730  mulsproplem5  27803  mulsproplem6  27804  mulsproplem7  27805  mulsproplem8  27806