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Theorem onint 7778
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 7764 . . . 4 Ord On
2 tz7.5 6386 . . . 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 570 . . . . . . . . . . . . . . 15 ((𝐴 ⊆ On ∧ 𝑥𝐴) → (𝐴 ⊆ On ∧ 𝑥 ∈ On))
6 ssel 3976 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ⊆ On → (𝑧𝐴𝑧 ∈ On))
7 ontri1 6399 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (𝑥𝑧 ↔ ¬ 𝑧𝑥))
8 ssel 3976 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝑧 → (𝑦𝑥𝑦𝑧))
97, 8syl6bir 254 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (¬ 𝑧𝑥 → (𝑦𝑥𝑦𝑧)))
109ex 414 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → (𝑧 ∈ On → (¬ 𝑧𝑥 → (𝑦𝑥𝑦𝑧))))
116, 10sylan9 509 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ⊆ On ∧ 𝑥 ∈ On) → (𝑧𝐴 → (¬ 𝑧𝑥 → (𝑦𝑥𝑦𝑧))))
1211com4r 94 . . . . . . . . . . . . . . . . . 18 (𝑦𝑥 → ((𝐴 ⊆ On ∧ 𝑥 ∈ On) → (𝑧𝐴 → (¬ 𝑧𝑥𝑦𝑧))))
1312imp31 419 . . . . . . . . . . . . . . . . 17 (((𝑦𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) ∧ 𝑧𝐴) → (¬ 𝑧𝑥𝑦𝑧))
1413ralimdva 3168 . . . . . . . . . . . . . . . 16 ((𝑦𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) → (∀𝑧𝐴 ¬ 𝑧𝑥 → ∀𝑧𝐴 𝑦𝑧))
15 disj 4448 . . . . . . . . . . . . . . . 16 ((𝐴𝑥) = ∅ ↔ ∀𝑧𝐴 ¬ 𝑧𝑥)
16 vex 3479 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
1716elint2 4958 . . . . . . . . . . . . . . . 16 (𝑦 𝐴 ↔ ∀𝑧𝐴 𝑦𝑧)
1814, 15, 173imtr4g 296 . . . . . . . . . . . . . . 15 ((𝑦𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) → ((𝐴𝑥) = ∅ → 𝑦 𝐴))
195, 18sylan2 594 . . . . . . . . . . . . . 14 ((𝑦𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥𝐴)) → ((𝐴𝑥) = ∅ → 𝑦 𝐴))
2019exp32 422 . . . . . . . . . . . . 13 (𝑦𝑥 → (𝐴 ⊆ On → (𝑥𝐴 → ((𝐴𝑥) = ∅ → 𝑦 𝐴))))
2120com4l 92 . . . . . . . . . . . 12 (𝐴 ⊆ On → (𝑥𝐴 → ((𝐴𝑥) = ∅ → (𝑦𝑥𝑦 𝐴))))
2221imp32 420 . . . . . . . . . . 11 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → (𝑦𝑥𝑦 𝐴))
2322ssrdv 3989 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → 𝑥 𝐴)
24 intss1 4968 . . . . . . . . . . 11 (𝑥𝐴 𝐴𝑥)
2524ad2antrl 727 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → 𝐴𝑥)
2623, 25eqssd 4000 . . . . . . . . 9 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → 𝑥 = 𝐴)
2726eleq1d 2819 . . . . . . . 8 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → (𝑥𝐴 𝐴𝐴))
2827biimpd 228 . . . . . . 7 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ (𝐴𝑥) = ∅)) → (𝑥𝐴 𝐴𝐴))
2928exp32 422 . . . . . 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 3154 . . 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 397   = wceq 1542  wcel 2107  wne 2941  wral 3062  wrex 3071  cin 3948  wss 3949  c0 4323   cint 4951  Ord word 6364  Oncon0 6365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369
This theorem is referenced by:  onint0  7779  onssmin  7780  onminesb  7781  onminsb  7782  oninton  7783  oneqmin  7788  oeeulem  8601  nnawordex  8637  unblem1  9295  unblem2  9296  tz9.12lem3  9784  scott0  9881  cardid2  9948  ackbij1lem18  10232  cardcf  10247  cff1  10253  cflim2  10258  cfss  10260  cofsmo  10264  fin23lem26  10320  pwfseqlem3  10655  gruina  10813  2ndcdisj  22960  sltval2  27159  nocvxmin  27280  lrrecfr  27427  rankeq1o  35143  dnnumch3  41789  oninfint  41985  inaex  43056
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