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Theorem onnminsb 7789
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 3683 . . . 4 (𝐴 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝐴 ∈ On ∧ 𝜓))
3 ssrab2 4077 . . . . 5 {𝑥 ∈ On ∣ 𝜑} ⊆ On
4 onnmin 7788 . . . . 5 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ 𝐴 ∈ {𝑥 ∈ On ∣ 𝜑}) → ¬ 𝐴 {𝑥 ∈ On ∣ 𝜑})
53, 4mpan 688 . . . 4 (𝐴 ∈ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝐴 {𝑥 ∈ On ∣ 𝜑})
62, 5sylbir 234 . . 3 ((𝐴 ∈ On ∧ 𝜓) → ¬ 𝐴 {𝑥 ∈ On ∣ 𝜑})
76ex 413 . 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 205  wa 396   = wceq 1541  wcel 2106  {crab 3432  wss 3948   cint 4950  Oncon0 6364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368
This theorem is referenced by:  onminex  7792  oawordeulem  8556  oeeulem  8603  nnawordex  8639  tcrank  9881  alephnbtwn  10068  cardaleph  10086  cardmin  10561  sltval2  27383  nosepeq  27412  nosupbnd2lem1  27442  noinfbnd2lem1  27457  naddwordnexlem4  42454
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