Theorem ordunifi 9326

Description: The maximum of a finite collection of ordinals is in the set. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)

Assertion

Ref	Expression
ordunifi	⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴)

Proof of Theorem ordunifi

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		epweon 7795	. . . . . 6 ⊢ E We On
2		weso 5676	. . . . . 6 ⊢ ( E We On → E Or On)
3	1, 2	ax-mp 5	. . . . 5 ⊢ E Or On
4		soss 5612	. . . . 5 ⊢ (𝐴 ⊆ On → ( E Or On → E Or 𝐴))
5	3, 4	mpi 20	. . . 4 ⊢ (𝐴 ⊆ On → E Or 𝐴)
6		fimax2g 9322	. . . 4 ⊢ (( E Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦)
7	5, 6	syl3an1 1164	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦)
8		ssel2 3978	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
9	8	adantlr 715	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
10		ssel2 3978	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
11	10	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ On)
12		epel 5587	. . . . . . . . . 10 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
13	12	notbii 320	. . . . . . . . 9 ⊢ (¬ 𝑥 E 𝑦 ↔ ¬ 𝑥 ∈ 𝑦)
14		ontri1 6418	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
15	13, 14	bitr4id 290	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥 E 𝑦 ↔ 𝑦 ⊆ 𝑥))
16	9, 11, 15	syl2anc 584	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑥 E 𝑦 ↔ 𝑦 ⊆ 𝑥))
17	16	ralbidva 3176	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥))
18		unissb 4939	. . . . . 6 ⊢ (∪ 𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)
19	17, 18	bitr4di 289	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∪ 𝐴 ⊆ 𝑥))
20	19	rexbidva 3177	. . . 4 ⊢ (𝐴 ⊆ On → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∃𝑥 ∈ 𝐴 ∪ 𝐴 ⊆ 𝑥))
21	20	3ad2ant1 1134	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∃𝑥 ∈ 𝐴 ∪ 𝐴 ⊆ 𝑥))
22	7, 21	mpbid 232	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∪ 𝐴 ⊆ 𝑥)
23		elssuni 4937	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
24		eqss 3999	. . . . 5 ⊢ (𝑥 = ∪ 𝐴 ↔ (𝑥 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑥))
25		eleq1 2829	. . . . . 6 ⊢ (𝑥 = ∪ 𝐴 → (𝑥 ∈ 𝐴 ↔ ∪ 𝐴 ∈ 𝐴))
26	25	biimpcd 249	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = ∪ 𝐴 → ∪ 𝐴 ∈ 𝐴))
27	24, 26	biimtrrid 243	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑥) → ∪ 𝐴 ∈ 𝐴))
28	23, 27	mpand 695	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∪ 𝐴 ⊆ 𝑥 → ∪ 𝐴 ∈ 𝐴))
29	28	rexlimiv 3148	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∪ 𝐴 ⊆ 𝑥 → ∪ 𝐴 ∈ 𝐴)
30	22, 29	syl 17	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951  ∅c0 4333  ∪ cuni 4907   class class class wbr 5143   E cep 5583   Or wor 5591   We wwe 5636  Oncon0 6384  Fincfn 8985

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-en 8986  df-fin 8989

This theorem is referenced by:  nnunifi  9327  oemapvali  9724  ttukeylem6  10554  limsucncmpi  36446  onfisupcl  43262  onsucunifi  43383