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Theorem onint 7811
Description: The intersection (infimum) of a nonempty class of ordinal numbers belongs to the class. Compare Exercise 4 of [TakeutiZaring] p. 45. (Contributed by NM, 31-Jan-1997.)
Assertion
Ref Expression
onint ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴𝐴)

Proof of Theorem onint
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordon 7798 . . . 4 Ord On
2 tz7.5 6404 . . . 4 ((Ord On ∧ 𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴 (𝐴𝑥) = ∅)
31, 2mp3an1 1449 . . 3 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴 (𝐴𝑥) = ∅)
4 ssel 3976 . . . . . . . . . . . . . . . 16 (𝐴 ⊆ On → (𝑥𝐴𝑥 ∈ On))
54imdistani 568 . . . . . . . . . . . . . . 15 ((𝐴 ⊆ On ∧ 𝑥𝐴) → (𝐴 ⊆ On ∧ 𝑥 ∈ On))
6 ssel 3976 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ⊆ On → (𝑧𝐴𝑧 ∈ On))
7 ontri1 6417 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (𝑥𝑧 ↔ ¬ 𝑧𝑥))
8 ssel 3976 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝑧 → (𝑦𝑥𝑦𝑧))
97, 8biimtrrdi 254 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (¬ 𝑧𝑥 → (𝑦𝑥𝑦𝑧)))
109ex 412 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → (𝑧 ∈ On → (¬ 𝑧𝑥 → (𝑦𝑥𝑦𝑧))))
116, 10sylan9 507 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ⊆ On ∧ 𝑥 ∈ On) → (𝑧𝐴 → (¬ 𝑧𝑥 → (𝑦𝑥𝑦𝑧))))
1211com4r 94 . . . . . . . . . . . . . . . . . 18 (𝑦𝑥 → ((𝐴 ⊆ On ∧ 𝑥 ∈ On) → (𝑧𝐴 → (¬ 𝑧𝑥𝑦𝑧))))
1312imp31 417 . . . . . . . . . . . . . . . . 17 (((𝑦𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) ∧ 𝑧𝐴) → (¬ 𝑧𝑥𝑦𝑧))
1413ralimdva 3166 . . . . . . . . . . . . . . . 16 ((𝑦𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) → (∀𝑧𝐴 ¬ 𝑧𝑥 → ∀𝑧𝐴 𝑦𝑧))
15 disj 4449 . . . . . . . . . . . . . . . 16 ((𝐴𝑥) = ∅ ↔ ∀𝑧𝐴 ¬ 𝑧𝑥)
16 vex 3483 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
1716elint2 4952 . . . . . . . . . . . . . . . 16 (𝑦 𝐴 ↔ ∀𝑧𝐴 𝑦𝑧)
1814, 15, 173imtr4g 296 . . . . . . . . . . . . . . 15 ((𝑦𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) → ((𝐴𝑥) = ∅ → 𝑦 𝐴))
195, 18sylan2 593 . . . . . . . . . . . . . 14 ((𝑦𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥𝐴)) → ((𝐴𝑥) = ∅ → 𝑦 𝐴))
2019exp32 420 . . . . . . . . . . . . 13 (𝑦𝑥 → (𝐴 ⊆ On → (𝑥𝐴 → ((𝐴𝑥) = ∅ → 𝑦 𝐴))))
2120com4l 92 . . . . . . . . . . . 12 (𝐴 ⊆ On → (𝑥𝐴 → ((𝐴𝑥) = ∅ → (𝑦𝑥𝑦 𝐴))))
2221imp32 418 . . . . . . . . . . 11 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → (𝑦𝑥𝑦 𝐴))
2322ssrdv 3988 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → 𝑥 𝐴)
24 intss1 4962 . . . . . . . . . . 11 (𝑥𝐴 𝐴𝑥)
2524ad2antrl 728 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → 𝐴𝑥)
2623, 25eqssd 4000 . . . . . . . . 9 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → 𝑥 = 𝐴)
2726eleq1d 2825 . . . . . . . 8 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → (𝑥𝐴 𝐴𝐴))
2827biimpd 229 . . . . . . 7 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → (𝑥𝐴 𝐴𝐴))
2928exp32 420 . . . . . 6 (𝐴 ⊆ On → (𝑥𝐴 → ((𝐴𝑥) = ∅ → (𝑥𝐴 𝐴𝐴))))
3029com34 91 . . . . 5 (𝐴 ⊆ On → (𝑥𝐴 → (𝑥𝐴 → ((𝐴𝑥) = ∅ → 𝐴𝐴))))
3130pm2.43d 53 . . . 4 (𝐴 ⊆ On → (𝑥𝐴 → ((𝐴𝑥) = ∅ → 𝐴𝐴)))
3231rexlimdv 3152 . . 3 (𝐴 ⊆ On → (∃𝑥𝐴 (𝐴𝑥) = ∅ → 𝐴𝐴))
333, 32syl5 34 . 2 (𝐴 ⊆ On → ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴𝐴))
3433anabsi5 669 1 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1539  wcel 2107  wne 2939  wral 3060  wrex 3069  cin 3949  wss 3950  c0 4332   cint 4945  Ord word 6382  Oncon0 6383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387
This theorem is referenced by:  onint0  7812  onssmin  7813  onminesb  7814  onminsb  7815  oninton  7816  oneqmin  7821  oeeulem  8640  nnawordex  8676  unblem1  9329  unblem2  9330  tz9.12lem3  9830  scott0  9927  cardid2  9994  ackbij1lem18  10277  cardcf  10293  cff1  10299  cflim2  10304  cfss  10306  cofsmo  10310  fin23lem26  10366  pwfseqlem3  10701  gruina  10859  2ndcdisj  23465  sltval2  27702  nocvxmin  27824  lrrecfr  27977  rankeq1o  36173  dnnumch3  43064  oninfint  43253  inaex  44321
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