Theorem onminsb 7840

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 4342	. . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
2		ssrab2 4008	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
3		onint 7836	. . . 4 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
4	2, 3	mpan 703	. . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
5	1, 4	sylbir 238	. 2 ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
6		nfrab1 3435	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝜑}
7	6	nfint 4920	. . . 4 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝜑}
8		nfcv 2926	. . . 4 ⊢ Ⅎ𝑥On
9		onminsb.1	. . . 4 ⊢ Ⅎ𝑥𝜓
10		onminsb.2	. . . 4 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝜑} → (𝜑 ↔ 𝜓))
11	7, 8, 9, 10	elrabf 3645	. . 3 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ On ∧ 𝜓))
12	11	simprbi 503	. 2 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} → 𝜓)
13	5, 12	syl 18	1 ⊢ (∃𝑥 ∈ On 𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090  {crab 3416   ⊆ wss 3900  ∅c0 4281  ∩ cint 4910  Oncon0 6386

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5172  df-tr 5217  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-ord 6389  df-on 6390

This theorem is used by:  oawordeulem  8598  rankidb  9843  cardmin2  10117  cardaleph  10205  cardmin  10685  onvf1odlem2  36066  onprcf1acwevdlem2  36080  vonf1oonfo  36083  onprcf1of1ord  36089  naddwordnexlem4  44658