Theorem ordunifi 9190

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 7718	. . . . . 6 ⊢ E We On
2		weso 5609	. . . . . 6 ⊢ ( E We On → E Or On)
3	1, 2	ax-mp 5	. . . . 5 ⊢ E Or On
4		soss 5546	. . . . 5 ⊢ (𝐴 ⊆ On → ( E Or On → E Or 𝐴))
5	3, 4	mpi 20	. . . 4 ⊢ (𝐴 ⊆ On → E Or 𝐴)
6		fimax2g 9186	. . . 4 ⊢ (( E Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦)
7	5, 6	syl3an1 1169	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦)
8		ssel2 3910	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
9	8	adantlr 721	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
10		ssel2 3910	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
11	10	adantr 481	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ On)
12		epel 5521	. . . . . . . . . 10 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
13	12	notbii 321	. . . . . . . . 9 ⊢ (¬ 𝑥 E 𝑦 ↔ ¬ 𝑥 ∈ 𝑦)
14		ontri1 6344	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
15	13, 14	bitr4id 291	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥 E 𝑦 ↔ 𝑦 ⊆ 𝑥))
16	9, 11, 15	syl2anc 590	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑥 E 𝑦 ↔ 𝑦 ⊆ 𝑥))
17	16	ralbidva 3160	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥))
18		unissb 4871	. . . . . 6 ⊢ (∪ 𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)
19	17, 18	bitr4di 290	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∪ 𝐴 ⊆ 𝑥))
20	19	rexbidva 3161	. . . 4 ⊢ (𝐴 ⊆ On → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∃𝑥 ∈ 𝐴 ∪ 𝐴 ⊆ 𝑥))
21	20	3ad2ant1 1139	. . 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 4869	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
24		eqss 3930	. . . . 5 ⊢ (𝑥 = ∪ 𝐴 ↔ (𝑥 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑥))
25		eleq1 2827	. . . . . 6 ⊢ (𝑥 = ∪ 𝐴 → (𝑥 ∈ 𝐴 ↔ ∪ 𝐴 ∈ 𝐴))
26	25	biimpcd 250	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = ∪ 𝐴 → ∪ 𝐴 ∈ 𝐴))
27	24, 26	biimtrrid 244	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑥) → ∪ 𝐴 ∈ 𝐴))
28	23, 27	mpand 701	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∪ 𝐴 ⊆ 𝑥 → ∪ 𝐴 ∈ 𝐴))
29	28	rexlimiv 3133	. 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 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  ∃wrex 3063   ⊆ wss 3883  ∅c0 4261  ∪ cuni 4838   class class class wbr 5072   E cep 5517   Or wor 5525   We wwe 5570  Oncon0 6310  Fincfn 8883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-om 7807  df-en 8884  df-fin 8887

This theorem is referenced by:  nnunifi  9191  oemapvali  9596  ttukeylem6  10427  fissorduni  35271  limsucncmpi  36673  onfisupcl  43695  onsucunifi  43815