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Theorem onnminsb 7524
 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 3604 . . . 4 (𝐴 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝐴 ∈ On ∧ 𝜓))
3 ssrab2 3986 . . . . 5 {𝑥 ∈ On ∣ 𝜑} ⊆ On
4 onnmin 7523 . . . . 5 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ 𝐴 ∈ {𝑥 ∈ On ∣ 𝜑}) → ¬ 𝐴 {𝑥 ∈ On ∣ 𝜑})
53, 4mpan 689 . . . 4 (𝐴 ∈ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝐴 {𝑥 ∈ On ∣ 𝜑})
62, 5sylbir 238 . . 3 ((𝐴 ∈ On ∧ 𝜓) → ¬ 𝐴 {𝑥 ∈ On ∣ 𝜑})
76ex 416 . 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3074   ⊆ wss 3860  ∩ cint 4841  Oncon0 6174 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-br 5037  df-opab 5099  df-tr 5143  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-ord 6177  df-on 6178 This theorem is referenced by:  onminex  7527  oawordeulem  8196  oeeulem  8243  nnawordex  8279  tcrank  9359  alephnbtwn  9544  cardaleph  9562  cardmin  10037  sltval2  33456  nosepeq  33485  nosupbnd2lem1  33515  noinfbnd2lem1  33530
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