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Theorem onnminsb 7508
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 3677 . . . 4 (𝐴 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝐴 ∈ On ∧ 𝜓))
3 ssrab2 4053 . . . . 5 {𝑥 ∈ On ∣ 𝜑} ⊆ On
4 onnmin 7507 . . . . 5 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ 𝐴 ∈ {𝑥 ∈ On ∣ 𝜑}) → ¬ 𝐴 {𝑥 ∈ On ∣ 𝜑})
53, 4mpan 686 . . . 4 (𝐴 ∈ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝐴 {𝑥 ∈ On ∣ 𝜑})
62, 5sylbir 236 . . 3 ((𝐴 ∈ On ∧ 𝜓) → ¬ 𝐴 {𝑥 ∈ On ∣ 𝜑})
76ex 413 . 2 (𝐴 ∈ On → (𝜓 → ¬ 𝐴 {𝑥 ∈ On ∣ 𝜑}))
87con2d 136 1 (𝐴 ∈ On → (𝐴 {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  {crab 3139  wss 3933   cint 4867  Oncon0 6184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188
This theorem is referenced by:  onminex  7511  oawordeulem  8169  oeeulem  8216  nnawordex  8252  tcrank  9301  alephnbtwn  9485  cardaleph  9503  cardmin  9974  sltval2  33060  nosepeq  33086  nosupbnd2lem1  33112
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