Theorem ordunifi 8614

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 7353	. . . . . 6 ⊢ E We On
2		weso 5434	. . . . . 6 ⊢ ( E We On → E Or On)
3	1, 2	ax-mp 5	. . . . 5 ⊢ E Or On
4		soss 5381	. . . . 5 ⊢ (𝐴 ⊆ On → ( E Or On → E Or 𝐴))
5	3, 4	mpi 20	. . . 4 ⊢ (𝐴 ⊆ On → E Or 𝐴)
6		fimax2g 8610	. . . 4 ⊢ (( E Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦)
7	5, 6	syl3an1 1156	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦)
8		ssel2 3884	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
9	8	adantlr 711	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
10		ssel2 3884	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
11	10	adantr 481	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ On)
12		ontri1 6100	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
13		epel 5357	. . . . . . . . . 10 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
14	13	notbii 321	. . . . . . . . 9 ⊢ (¬ 𝑥 E 𝑦 ↔ ¬ 𝑥 ∈ 𝑦)
15	12, 14	syl6rbbr 291	. . . . . . . 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 3163	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥))
18		unissb 4776	. . . . . 6 ⊢ (∪ 𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)
19	17, 18	syl6bbr 290	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∪ 𝐴 ⊆ 𝑥))
20	19	rexbidva 3259	. . . 4 ⊢ (𝐴 ⊆ On → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∃𝑥 ∈ 𝐴 ∪ 𝐴 ⊆ 𝑥))
21	20	3ad2ant1 1126	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∃𝑥 ∈ 𝐴 ∪ 𝐴 ⊆ 𝑥))
22	7, 21	mpbid 233	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∪ 𝐴 ⊆ 𝑥)
23		elssuni 4774	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
24		eqss 3904	. . . . 5 ⊢ (𝑥 = ∪ 𝐴 ↔ (𝑥 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑥))
25		eleq1 2870	. . . . . 6 ⊢ (𝑥 = ∪ 𝐴 → (𝑥 ∈ 𝐴 ↔ ∪ 𝐴 ∈ 𝐴))
26	25	biimpcd 250	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = ∪ 𝐴 → ∪ 𝐴 ∈ 𝐴))
27	24, 26	syl5bir 244	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑥) → ∪ 𝐴 ∈ 𝐴))
28	23, 27	mpand 691	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∪ 𝐴 ⊆ 𝑥 → ∪ 𝐴 ∈ 𝐴))
29	28	rexlimiv 3243	. 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 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081   ≠ wne 2984  ∀wral 3105  ∃wrex 3106   ⊆ wss 3859  ∅c0 4211  ∪ cuni 4745   class class class wbr 4962   E cep 5352   Or wor 5361   We wwe 5401  Oncon0 6066  Fincfn 8357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-om 7437  df-1o 7953  df-er 8139  df-en 8358  df-fin 8361

This theorem is referenced by:  nnunifi  8615  oemapvali  8993  ttukeylem6  9782  limsucncmpi  33403