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Theorem onint 6887
 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 6874 . . . 4 Ord On
2 tz7.5 5661 . . . 4 ((Ord On ∧ 𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴 (𝐴𝑥) = ∅)
31, 2mp3an1 1403 . . 3 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴 (𝐴𝑥) = ∅)
4 ssel 3562 . . . . . . . . . . . . . . . 16 (𝐴 ⊆ On → (𝑥𝐴𝑥 ∈ On))
54imdistani 722 . . . . . . . . . . . . . . 15 ((𝐴 ⊆ On ∧ 𝑥𝐴) → (𝐴 ⊆ On ∧ 𝑥 ∈ On))
6 ssel 3562 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ⊆ On → (𝑧𝐴𝑧 ∈ On))
7 ontri1 5674 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (𝑥𝑧 ↔ ¬ 𝑧𝑥))
8 ssel 3562 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝑧 → (𝑦𝑥𝑦𝑧))
97, 8syl6bir 243 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (¬ 𝑧𝑥 → (𝑦𝑥𝑦𝑧)))
109ex 449 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → (𝑧 ∈ On → (¬ 𝑧𝑥 → (𝑦𝑥𝑦𝑧))))
116, 10sylan9 687 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ⊆ On ∧ 𝑥 ∈ On) → (𝑧𝐴 → (¬ 𝑧𝑥 → (𝑦𝑥𝑦𝑧))))
1211com4r 92 . . . . . . . . . . . . . . . . . 18 (𝑦𝑥 → ((𝐴 ⊆ On ∧ 𝑥 ∈ On) → (𝑧𝐴 → (¬ 𝑧𝑥𝑦𝑧))))
1312imp31 447 . . . . . . . . . . . . . . . . 17 (((𝑦𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) ∧ 𝑧𝐴) → (¬ 𝑧𝑥𝑦𝑧))
1413ralimdva 2945 . . . . . . . . . . . . . . . 16 ((𝑦𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) → (∀𝑧𝐴 ¬ 𝑧𝑥 → ∀𝑧𝐴 𝑦𝑧))
15 disj 3969 . . . . . . . . . . . . . . . 16 ((𝐴𝑥) = ∅ ↔ ∀𝑧𝐴 ¬ 𝑧𝑥)
16 vex 3176 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
1716elint2 4417 . . . . . . . . . . . . . . . 16 (𝑦 𝐴 ↔ ∀𝑧𝐴 𝑦𝑧)
1814, 15, 173imtr4g 284 . . . . . . . . . . . . . . 15 ((𝑦𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) → ((𝐴𝑥) = ∅ → 𝑦 𝐴))
195, 18sylan2 490 . . . . . . . . . . . . . 14 ((𝑦𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥𝐴)) → ((𝐴𝑥) = ∅ → 𝑦 𝐴))
2019exp32 629 . . . . . . . . . . . . 13 (𝑦𝑥 → (𝐴 ⊆ On → (𝑥𝐴 → ((𝐴𝑥) = ∅ → 𝑦 𝐴))))
2120com4l 90 . . . . . . . . . . . 12 (𝐴 ⊆ On → (𝑥𝐴 → ((𝐴𝑥) = ∅ → (𝑦𝑥𝑦 𝐴))))
2221imp32 448 . . . . . . . . . . 11 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → (𝑦𝑥𝑦 𝐴))
2322ssrdv 3574 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → 𝑥 𝐴)
24 intss1 4427 . . . . . . . . . . 11 (𝑥𝐴 𝐴𝑥)
2524ad2antrl 760 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → 𝐴𝑥)
2623, 25eqssd 3585 . . . . . . . . 9 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → 𝑥 = 𝐴)
2726eleq1d 2672 . . . . . . . 8 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → (𝑥𝐴 𝐴𝐴))
2827biimpd 218 . . . . . . 7 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → (𝑥𝐴 𝐴𝐴))
2928exp32 629 . . . . . 6 (𝐴 ⊆ On → (𝑥𝐴 → ((𝐴𝑥) = ∅ → (𝑥𝐴 𝐴𝐴))))
3029com34 89 . . . . 5 (𝐴 ⊆ On → (𝑥𝐴 → (𝑥𝐴 → ((𝐴𝑥) = ∅ → 𝐴𝐴))))
3130pm2.43d 51 . . . 4 (𝐴 ⊆ On → (𝑥𝐴 → ((𝐴𝑥) = ∅ → 𝐴𝐴)))
3231rexlimdv 3012 . . 3 (𝐴 ⊆ On → (∃𝑥𝐴 (𝐴𝑥) = ∅ → 𝐴𝐴))
333, 32syl5 33 . 2 (𝐴 ⊆ On → ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴𝐴))
3433anabsi5 854 1 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ∩ cint 4410  Ord word 5639  Oncon0 5640 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644 This theorem is referenced by:  onint0  6888  onssmin  6889  onminesb  6890  onminsb  6891  oninton  6892  oneqmin  6897  oeeulem  7568  nnawordex  7604  unblem1  8097  unblem2  8098  tz9.12lem3  8535  scott0  8632  cardid2  8662  ackbij1lem18  8942  cardcf  8957  cff1  8963  cflim2  8968  cfss  8970  cofsmo  8974  fin23lem26  9030  pwfseqlem3  9361  gruina  9519  2ndcdisj  21069  sltval2  31053  nocvxmin  31090  nobndlem5  31095  rankeq1o  31448  dnnumch3  36635
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