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Theorem onminsb 7786
Description: If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses implicit substitution. Theorem Schema 62 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.)
Hypotheses
Ref Expression
onminsb.1 𝑥𝜓
onminsb.2 (𝑥 = {𝑥 ∈ On ∣ 𝜑} → (𝜑𝜓))
Assertion
Ref Expression
onminsb (∃𝑥 ∈ On 𝜑𝜓)

Proof of Theorem onminsb
StepHypRef Expression
1 rabn0 4385 . . 3 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
2 ssrab2 4077 . . . 4 {𝑥 ∈ On ∣ 𝜑} ⊆ On
3 onint 7782 . . . 4 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
42, 3mpan 687 . . 3 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ → {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
51, 4sylbir 234 . 2 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
6 nfrab1 3450 . . . . 5 𝑥{𝑥 ∈ On ∣ 𝜑}
76nfint 4960 . . . 4 𝑥 {𝑥 ∈ On ∣ 𝜑}
8 nfcv 2902 . . . 4 𝑥On
9 onminsb.1 . . . 4 𝑥𝜓
10 onminsb.2 . . . 4 (𝑥 = {𝑥 ∈ On ∣ 𝜑} → (𝜑𝜓))
117, 8, 9, 10elrabf 3679 . . 3 ( {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} ↔ ( {𝑥 ∈ On ∣ 𝜑} ∈ On ∧ 𝜓))
1211simprbi 496 . 2 ( {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} → 𝜓)
135, 12syl 17 1 (∃𝑥 ∈ On 𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1540  wnf 1784  wcel 2105  wne 2939  wrex 3069  {crab 3431  wss 3948  c0 4322   cint 4950  Oncon0 6364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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:  oawordeulem  8560  rankidb  9801  cardmin2  10000  cardaleph  10090  cardmin  10565  naddwordnexlem4  42615
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