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Theorem onnminsb 6873
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 3330 . . . 4 (𝐴 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝐴 ∈ On ∧ 𝜓))
3 ssrab2 3649 . . . . 5 {𝑥 ∈ On ∣ 𝜑} ⊆ On
4 onnmin 6872 . . . . 5 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ 𝐴 ∈ {𝑥 ∈ On ∣ 𝜑}) → ¬ 𝐴 {𝑥 ∈ On ∣ 𝜑})
53, 4mpan 701 . . . 4 (𝐴 ∈ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝐴 {𝑥 ∈ On ∣ 𝜑})
62, 5sylbir 223 . . 3 ((𝐴 ∈ On ∧ 𝜓) → ¬ 𝐴 {𝑥 ∈ On ∣ 𝜑})
76ex 448 . 2 (𝐴 ∈ On → (𝜓 → ¬ 𝐴 {𝑥 ∈ On ∣ 𝜑}))
87con2d 127 1 (𝐴 ∈ On → (𝐴 {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  {crab 2899  wss 3539   cint 4404  Oncon0 5626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-br 4578  df-opab 4638  df-tr 4675  df-eprel 4939  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-ord 5629  df-on 5630
This theorem is referenced by:  onminex  6876  oawordeulem  7498  oeeulem  7545  nnawordex  7581  tcrank  8607  alephnbtwn  8754  cardaleph  8772  cardmin  9242  sltval2  30859
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