Theorem onintrab2im 4477

Description: An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.)

Assertion

Ref	Expression
onintrab2im	⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On)

Proof of Theorem onintrab2im

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3213	. 2 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
2		nfrab1 2636	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝜑}
3	2	nfcri 2293	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}
4	3	nfex 1617	. . 3 ⊢ Ⅎ𝑥∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}
5		rabid 2632	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑥 ∈ On ∧ 𝜑))
6		elex2 2728	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
7	5, 6	sylbir 134	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝜑) → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
8	7	ex 114	. . 3 ⊢ (𝑥 ∈ On → (𝜑 → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}))
9	4, 8	rexlimi 2567	. 2 ⊢ (∃𝑥 ∈ On 𝜑 → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
10		onintonm 4476	. 2 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}) → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On)
11	1, 9, 10	sylancr 411	1 ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On)

Syntax hints:   → wi 4   ∧ wa 103  ∃wex 1472   ∈ wcel 2128  ∃wrex 2436  {crab 2439   ⊆ wss 3102  ∩ cint 3807  Oncon0 4323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-int 3808  df-tr 4063  df-iord 4326  df-on 4328  df-suc 4331

This theorem is referenced by:  cardcl  7116