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Theorem onminsb 6868
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 3911 . . 3 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
2 ssrab2 3649 . . . 4 {𝑥 ∈ On ∣ 𝜑} ⊆ On
3 onint 6864 . . . 4 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
42, 3mpan 701 . . 3 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ → {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
51, 4sylbir 223 . 2 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
6 nfrab1 3098 . . . . 5 𝑥{𝑥 ∈ On ∣ 𝜑}
76nfint 4415 . . . 4 𝑥 {𝑥 ∈ On ∣ 𝜑}
8 nfcv 2750 . . . 4 𝑥On
9 onminsb.1 . . . 4 𝑥𝜓
10 onminsb.2 . . . 4 (𝑥 = {𝑥 ∈ On ∣ 𝜑} → (𝜑𝜓))
117, 8, 9, 10elrabf 3328 . . 3 ( {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} ↔ ( {𝑥 ∈ On ∣ 𝜑} ∈ On ∧ 𝜓))
1211simprbi 478 . 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 194   = wceq 1474  wnf 1698  wcel 1976  wne 2779  wrex 2896  {crab 2899  wss 3539  c0 3873   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 2032  ax-13 2232  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:  oawordeulem  7498  rankidb  8523  cardmin2  8684  cardaleph  8772  cardmin  9242
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