Theorem ordunifi 8994

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 7603	. . . . . 6 ⊢ E We On
2		weso 5571	. . . . . 6 ⊢ ( E We On → E Or On)
3	1, 2	ax-mp 5	. . . . 5 ⊢ E Or On
4		soss 5514	. . . . 5 ⊢ (𝐴 ⊆ On → ( E Or On → E Or 𝐴))
5	3, 4	mpi 20	. . . 4 ⊢ (𝐴 ⊆ On → E Or 𝐴)
6		fimax2g 8990	. . . 4 ⊢ (( E Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦)
7	5, 6	syl3an1 1161	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦)
8		ssel2 3912	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
9	8	adantlr 711	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
10		ssel2 3912	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
11	10	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ On)
12		epel 5489	. . . . . . . . . 10 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
13	12	notbii 319	. . . . . . . . 9 ⊢ (¬ 𝑥 E 𝑦 ↔ ¬ 𝑥 ∈ 𝑦)
14		ontri1 6285	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
15	13, 14	bitr4id 289	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥 E 𝑦 ↔ 𝑦 ⊆ 𝑥))
16	9, 11, 15	syl2anc 583	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑥 E 𝑦 ↔ 𝑦 ⊆ 𝑥))
17	16	ralbidva 3119	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥))
18		unissb 4870	. . . . . 6 ⊢ (∪ 𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)
19	17, 18	bitr4di 288	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∪ 𝐴 ⊆ 𝑥))
20	19	rexbidva 3224	. . . 4 ⊢ (𝐴 ⊆ On → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∃𝑥 ∈ 𝐴 ∪ 𝐴 ⊆ 𝑥))
21	20	3ad2ant1 1131	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∃𝑥 ∈ 𝐴 ∪ 𝐴 ⊆ 𝑥))
22	7, 21	mpbid 231	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∪ 𝐴 ⊆ 𝑥)
23		elssuni 4868	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
24		eqss 3932	. . . . 5 ⊢ (𝑥 = ∪ 𝐴 ↔ (𝑥 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑥))
25		eleq1 2826	. . . . . 6 ⊢ (𝑥 = ∪ 𝐴 → (𝑥 ∈ 𝐴 ↔ ∪ 𝐴 ∈ 𝐴))
26	25	biimpcd 248	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = ∪ 𝐴 → ∪ 𝐴 ∈ 𝐴))
27	24, 26	syl5bir 242	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑥) → ∪ 𝐴 ∈ 𝐴))
28	23, 27	mpand 691	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∪ 𝐴 ⊆ 𝑥 → ∪ 𝐴 ∈ 𝐴))
29	28	rexlimiv 3208	. 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 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883  ∅c0 4253  ∪ cuni 4836   class class class wbr 5070   E cep 5485   Or wor 5493   We wwe 5534  Oncon0 6251  Fincfn 8691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-en 8692  df-fin 8695

This theorem is referenced by:  nnunifi  8995  oemapvali  9372  ttukeylem6  10201  limsucncmpi  34561