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Theorem onminsb 7041
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 3991 . . 3 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
2 ssrab2 3720 . . . 4 {𝑥 ∈ On ∣ 𝜑} ⊆ On
3 onint 7037 . . . 4 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
42, 3mpan 706 . . 3 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ → {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
51, 4sylbir 225 . 2 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
6 nfrab1 3152 . . . . 5 𝑥{𝑥 ∈ On ∣ 𝜑}
76nfint 4518 . . . 4 𝑥 {𝑥 ∈ On ∣ 𝜑}
8 nfcv 2793 . . . 4 𝑥On
9 onminsb.1 . . . 4 𝑥𝜓
10 onminsb.2 . . . 4 (𝑥 = {𝑥 ∈ On ∣ 𝜑} → (𝜑𝜓))
117, 8, 9, 10elrabf 3392 . . 3 ( {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} ↔ ( {𝑥 ∈ On ∣ 𝜑} ∈ On ∧ 𝜓))
1211simprbi 479 . 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 196   = wceq 1523  wnf 1748  wcel 2030  wne 2823  wrex 2942  {crab 2945  wss 3607  c0 3948   cint 4507  Oncon0 5761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765
This theorem is referenced by:  oawordeulem  7679  rankidb  8701  cardmin2  8862  cardaleph  8950  cardmin  9424
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