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Theorem onnminsb 7820
Description: An ordinal number smaller than the minimum of a set of ordinal numbers does not have the property determining that set. 𝜓 is the wff resulting from the substitution of 𝐴 for 𝑥 in wff 𝜑. (Contributed by NM, 9-Nov-2003.)
Hypothesis
Ref Expression
onnminsb.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
onnminsb (𝐴 ∈ On → (𝐴 {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem onnminsb
StepHypRef Expression
1 onnminsb.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
21elrab 3691 . . . 4 (𝐴 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝐴 ∈ On ∧ 𝜓))
3 ssrab2 4079 . . . . 5 {𝑥 ∈ On ∣ 𝜑} ⊆ On
4 onnmin 7819 . . . . 5 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ 𝐴 ∈ {𝑥 ∈ On ∣ 𝜑}) → ¬ 𝐴 {𝑥 ∈ On ∣ 𝜑})
53, 4mpan 690 . . . 4 (𝐴 ∈ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝐴 {𝑥 ∈ On ∣ 𝜑})
62, 5sylbir 235 . . 3 ((𝐴 ∈ On ∧ 𝜓) → ¬ 𝐴 {𝑥 ∈ On ∣ 𝜑})
76ex 412 . 2 (𝐴 ∈ On → (𝜓 → ¬ 𝐴 {𝑥 ∈ On ∣ 𝜑}))
87con2d 134 1 (𝐴 ∈ On → (𝐴 {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  {crab 3435  wss 3950   cint 4945  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:  onminex  7823  oawordeulem  8593  oeeulem  8640  nnawordex  8676  tcrank  9925  alephnbtwn  10112  cardaleph  10130  cardmin  10605  sltval2  27702  nosepeq  27731  nosupbnd2lem1  27761  noinfbnd2lem1  27776  naddwordnexlem4  43419
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