Theorem onminsb 7508

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	⊢ Ⅎ𝑥𝜓
onminsb.2	⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝜑} → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
onminsb	⊢ (∃𝑥 ∈ On 𝜑 → 𝜓)

Proof of Theorem onminsb

Step	Hyp	Ref	Expression
1		rabn0 4338	. . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
2		ssrab2 4055	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
3		onint 7504	. . . 4 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
4	2, 3	mpan 688	. . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
5	1, 4	sylbir 237	. 2 ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
6		nfrab1 3384	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝜑}
7	6	nfint 4878	. . . 4 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝜑}
8		nfcv 2977	. . . 4 ⊢ Ⅎ𝑥On
9		onminsb.1	. . . 4 ⊢ Ⅎ𝑥𝜓
10		onminsb.2	. . . 4 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝜑} → (𝜑 ↔ 𝜓))
11	7, 8, 9, 10	elrabf 3675	. . 3 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ On ∧ 𝜓))
12	11	simprbi 499	. 2 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} → 𝜓)
13	5, 12	syl 17	1 ⊢ (∃𝑥 ∈ On 𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1533  Ⅎwnf 1780   ∈ wcel 2110   ≠ wne 3016  ∃wrex 3139  {crab 3142   ⊆ wss 3935  ∅c0 4290  ∩ cint 4868  Oncon0 6185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-br 5059  df-opab 5121  df-tr 5165  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-ord 6188  df-on 6189

This theorem is referenced by:  oawordeulem  8174  rankidb  9223  cardmin2  9421  cardaleph  9509  cardmin  9980