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Theorem onminsb 7734
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 4350 . . 3 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
2 ssrab2 4042 . . . 4 {𝑥 ∈ On ∣ 𝜑} ⊆ On
3 onint 7730 . . . 4 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
42, 3mpan 689 . . 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 3429 . . . . 5 𝑥{𝑥 ∈ On ∣ 𝜑}
76nfint 4922 . . . 4 𝑥 {𝑥 ∈ On ∣ 𝜑}
8 nfcv 2908 . . . 4 𝑥On
9 onminsb.1 . . . 4 𝑥𝜓
10 onminsb.2 . . . 4 (𝑥 = {𝑥 ∈ On ∣ 𝜑} → (𝜑𝜓))
117, 8, 9, 10elrabf 3646 . . 3 ( {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} ↔ ( {𝑥 ∈ On ∣ 𝜑} ∈ On ∧ 𝜓))
1211simprbi 498 . 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 1542  wnf 1786  wcel 2107  wne 2944  wrex 3074  {crab 3410  wss 3915  c0 4287   cint 4912  Oncon0 6322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326
This theorem is referenced by:  oawordeulem  8506  rankidb  9743  cardmin2  9942  cardaleph  10032  cardmin  10507  naddwordnexlem4  41747
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