Theorem onminesb 6998

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 3958	. . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
2		ssrab2 3687	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
3		onint 6995	. . . 4 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
4	2, 3	mpan 706	. . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
5	1, 4	sylbir 225	. 2 ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
6		nfcv 2764	. . . 4 ⊢ Ⅎ𝑥On
7	6	elrabsf 3474	. . 3 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ On ∧ [∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑))
8	7	simprbi 480	. 2 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} → [∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)
9	5, 8	syl 17	1 ⊢ (∃𝑥 ∈ On 𝜑 → [∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)

Syntax hints:   → wi 4   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913  {crab 2916  [wsbc 3435   ⊆ wss 3574  ∅c0 3915  ∩ cint 4475  Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  onminex  7007