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Theorem onint 7787
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 7773 . . . 4 Ord On
2 tz7.5 6384 . . . 4 ((Ord On ∧ 𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴 (𝐴𝑥) = ∅)
31, 2mp3an1 1445 . . 3 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴 (𝐴𝑥) = ∅)
4 ssel 3971 . . . . . . . . . . . . . . . 16 (𝐴 ⊆ On → (𝑥𝐴𝑥 ∈ On))
54imdistani 568 . . . . . . . . . . . . . . 15 ((𝐴 ⊆ On ∧ 𝑥𝐴) → (𝐴 ⊆ On ∧ 𝑥 ∈ On))
6 ssel 3971 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ⊆ On → (𝑧𝐴𝑧 ∈ On))
7 ontri1 6397 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (𝑥𝑧 ↔ ¬ 𝑧𝑥))
8 ssel 3971 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝑧 → (𝑦𝑥𝑦𝑧))
97, 8syl6bir 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 3162 . . . . . . . . . . . . . . . 16 ((𝑦𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) → (∀𝑧𝐴 ¬ 𝑧𝑥 → ∀𝑧𝐴 𝑦𝑧))
15 disj 4443 . . . . . . . . . . . . . . . 16 ((𝐴𝑥) = ∅ ↔ ∀𝑧𝐴 ¬ 𝑧𝑥)
16 vex 3473 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
1716elint2 4951 . . . . . . . . . . . . . . . 16 (𝑦 𝐴 ↔ ∀𝑧𝐴 𝑦𝑧)
1814, 15, 173imtr4g 296 . . . . . . . . . . . . . . 15 ((𝑦𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) → ((𝐴𝑥) = ∅ → 𝑦 𝐴))
195, 18sylan2 592 . . . . . . . . . . . . . 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 3984 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → 𝑥 𝐴)
24 intss1 4961 . . . . . . . . . . 11 (𝑥𝐴 𝐴𝑥)
2524ad2antrl 727 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → 𝐴𝑥)
2623, 25eqssd 3995 . . . . . . . . 9 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → 𝑥 = 𝐴)
2726eleq1d 2813 . . . . . . . 8 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → (𝑥𝐴 𝐴𝐴))
2827biimpd 228 . . . . . . 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 3148 . . 3 (𝐴 ⊆ On → (∃𝑥𝐴 (𝐴𝑥) = ∅ → 𝐴𝐴))
333, 32syl5 34 . 2 (𝐴 ⊆ On → ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴𝐴))
3433anabsi5 668 1 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1534  wcel 2099  wne 2935  wral 3056  wrex 3065  cin 3943  wss 3944  c0 4318   cint 4944  Ord word 6362  Oncon0 6363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-br 5143  df-opab 5205  df-tr 5260  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6366  df-on 6367
This theorem is referenced by:  onint0  7788  onssmin  7789  onminesb  7790  onminsb  7791  oninton  7792  oneqmin  7797  oeeulem  8615  nnawordex  8651  unblem1  9313  unblem2  9314  tz9.12lem3  9806  scott0  9903  cardid2  9970  ackbij1lem18  10254  cardcf  10269  cff1  10275  cflim2  10280  cfss  10282  cofsmo  10286  fin23lem26  10342  pwfseqlem3  10677  gruina  10835  2ndcdisj  23353  sltval2  27582  nocvxmin  27704  lrrecfr  27853  rankeq1o  35757  dnnumch3  42443  oninfint  42636  inaex  43706
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