Theorem onminesb 7643

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 4319	. . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
2		ssrab2 4013	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
3		onint 7640	. . . 4 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
4	2, 3	mpan 687	. . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
5	1, 4	sylbir 234	. 2 ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
6		nfcv 2907	. . . 4 ⊢ Ⅎ𝑥On
7	6	elrabsf 3764	. . 3 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ On ∧ [∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑))
8	7	simprbi 497	. 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 2106   ≠ wne 2943  ∃wrex 3065  {crab 3068  [wsbc 3716   ⊆ wss 3887  ∅c0 4256  ∩ cint 4879  Oncon0 6266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270

This theorem is referenced by:  onminex  7652