Theorem onminesb 7792

Description: If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses explicit substitution. Theorem Schema 62 of [Suppes] p. 228. (Contributed by NM, 29-Sep-2003.)

Assertion

Ref	Expression
onminesb	⊢ (∃𝑥 ∈ On 𝜑 → [∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)

Proof of Theorem onminesb

Step	Hyp	Ref	Expression
1		rabn0 4339	. . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
2		ssrab2 4028	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
3		onint 7789	. . . 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		nfcv 2922	. . . 4 ⊢ Ⅎ𝑥On
7	6	elrabsf 3784	. . 3 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ On ∧ [∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑))
8	7	simprbi 503	. 2 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} → [∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)
9	5, 8	syl 18	1 ⊢ (∃𝑥 ∈ On 𝜑 → [∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ≠ wne 2955  ∃wrex 3086  {crab 3412  [wsbc 3739   ⊆ wss 3899  ∅c0 4279  ∩ cint 4907  Oncon0 6357

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pr 5398

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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6360  df-on 6361

This theorem is used by:  onminex  7801