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Theorem ordunifi 8251
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 ≠ ∅) → 𝐴𝐴)

Proof of Theorem ordunifi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 epweon 7025 . . . . . 6 E We On
2 weso 5134 . . . . . 6 ( E We On → E Or On)
31, 2ax-mp 5 . . . . 5 E Or On
4 soss 5082 . . . . 5 (𝐴 ⊆ On → ( E Or On → E Or 𝐴))
53, 4mpi 20 . . . 4 (𝐴 ⊆ On → E Or 𝐴)
6 fimax2g 8247 . . . 4 (( E Or 𝐴𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 E 𝑦)
75, 6syl3an1 1399 . . 3 ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 E 𝑦)
8 ssel2 3631 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝑦𝐴) → 𝑦 ∈ On)
98adantlr 751 . . . . . . . 8 (((𝐴 ⊆ On ∧ 𝑥𝐴) ∧ 𝑦𝐴) → 𝑦 ∈ On)
10 ssel2 3631 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝑥𝐴) → 𝑥 ∈ On)
1110adantr 480 . . . . . . . 8 (((𝐴 ⊆ On ∧ 𝑥𝐴) ∧ 𝑦𝐴) → 𝑥 ∈ On)
12 ontri1 5795 . . . . . . . . 9 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥 ↔ ¬ 𝑥𝑦))
13 epel 5061 . . . . . . . . . 10 (𝑥 E 𝑦𝑥𝑦)
1413notbii 309 . . . . . . . . 9 𝑥 E 𝑦 ↔ ¬ 𝑥𝑦)
1512, 14syl6rbbr 279 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥 E 𝑦𝑦𝑥))
169, 11, 15syl2anc 694 . . . . . . 7 (((𝐴 ⊆ On ∧ 𝑥𝐴) ∧ 𝑦𝐴) → (¬ 𝑥 E 𝑦𝑦𝑥))
1716ralbidva 3014 . . . . . 6 ((𝐴 ⊆ On ∧ 𝑥𝐴) → (∀𝑦𝐴 ¬ 𝑥 E 𝑦 ↔ ∀𝑦𝐴 𝑦𝑥))
18 unissb 4501 . . . . . 6 ( 𝐴𝑥 ↔ ∀𝑦𝐴 𝑦𝑥)
1917, 18syl6bbr 278 . . . . 5 ((𝐴 ⊆ On ∧ 𝑥𝐴) → (∀𝑦𝐴 ¬ 𝑥 E 𝑦 𝐴𝑥))
2019rexbidva 3078 . . . 4 (𝐴 ⊆ On → (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 E 𝑦 ↔ ∃𝑥𝐴 𝐴𝑥))
21203ad2ant1 1102 . . 3 ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 E 𝑦 ↔ ∃𝑥𝐴 𝐴𝑥))
227, 21mpbid 222 . 2 ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴 𝐴𝑥)
23 elssuni 4499 . . . 4 (𝑥𝐴𝑥 𝐴)
24 eqss 3651 . . . . 5 (𝑥 = 𝐴 ↔ (𝑥 𝐴 𝐴𝑥))
25 eleq1 2718 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐴 𝐴𝐴))
2625biimpcd 239 . . . . 5 (𝑥𝐴 → (𝑥 = 𝐴 𝐴𝐴))
2724, 26syl5bir 233 . . . 4 (𝑥𝐴 → ((𝑥 𝐴 𝐴𝑥) → 𝐴𝐴))
2823, 27mpand 711 . . 3 (𝑥𝐴 → ( 𝐴𝑥 𝐴𝐴))
2928rexlimiv 3056 . 2 (∃𝑥𝐴 𝐴𝑥 𝐴𝐴)
3022, 29syl 17 1 ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  wss 3607  c0 3948   cuni 4468   class class class wbr 4685   E cep 5057   Or wor 5063   We wwe 5101  Oncon0 5761  Fincfn 7997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-er 7787  df-en 7998  df-fin 8001
This theorem is referenced by:  nnunifi  8252  oemapvali  8619  ttukeylem6  9374  limsucncmpi  32569
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