Theorem ordunifi 9202

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 7730	. . . . . 6 ⊢ E We On
2		weso 5623	. . . . . 6 ⊢ ( E We On → E Or On)
3	1, 2	ax-mp 5	. . . . 5 ⊢ E Or On
4		soss 5560	. . . . 5 ⊢ (𝐴 ⊆ On → ( E Or On → E Or 𝐴))
5	3, 4	mpi 20	. . . 4 ⊢ (𝐴 ⊆ On → E Or 𝐴)
6		fimax2g 9198	. . . 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 3930	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
9	8	adantlr 716	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
10		ssel2 3930	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
11	10	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ On)
12		epel 5535	. . . . . . . . . 10 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
13	12	notbii 320	. . . . . . . . 9 ⊢ (¬ 𝑥 E 𝑦 ↔ ¬ 𝑥 ∈ 𝑦)
14		ontri1 6359	. . . . . . . . 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 585	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑥 E 𝑦 ↔ 𝑦 ⊆ 𝑥))
17	16	ralbidva 3159	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥))
18		unissb 4898	. . . . . 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 3160	. . . 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 4896	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
24		eqss 3951	. . . . 5 ⊢ (𝑥 = ∪ 𝐴 ↔ (𝑥 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑥))
25		eleq1 2825	. . . . . 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 696	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∪ 𝐴 ⊆ 𝑥 → ∪ 𝐴 ∈ 𝐴))
29	28	rexlimiv 3132	. 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 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062   ⊆ wss 3903  ∅c0 4287  ∪ cuni 4865   class class class wbr 5100   E cep 5531   Or wor 5539   We wwe 5584  Oncon0 6325  Fincfn 8895

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7819  df-en 8896  df-fin 8899

This theorem is referenced by:  nnunifi  9203  oemapvali  9605  ttukeylem6  10436  fissorduni  35271  limsucncmpi  36665  onfisupcl  43611  onsucunifi  43731