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Theorem onminsb 7814
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 4389 . . 3 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
2 ssrab2 4080 . . . 4 {𝑥 ∈ On ∣ 𝜑} ⊆ On
3 onint 7810 . . . 4 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
42, 3mpan 690 . . 3 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ → {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
51, 4sylbir 235 . 2 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
6 nfrab1 3457 . . . . 5 𝑥{𝑥 ∈ On ∣ 𝜑}
76nfint 4956 . . . 4 𝑥 {𝑥 ∈ On ∣ 𝜑}
8 nfcv 2905 . . . 4 𝑥On
9 onminsb.1 . . . 4 𝑥𝜓
10 onminsb.2 . . . 4 (𝑥 = {𝑥 ∈ On ∣ 𝜑} → (𝜑𝜓))
117, 8, 9, 10elrabf 3688 . . 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 206   = wceq 1540  wnf 1783  wcel 2108  wne 2940  wrex 3070  {crab 3436  wss 3951  c0 4333   cint 4946  Oncon0 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388
This theorem is referenced by:  oawordeulem  8592  rankidb  9840  cardmin2  10039  cardaleph  10129  cardmin  10604  naddwordnexlem4  43414
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