Theorem onintrab2im 4587

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 3289	. 2 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
2		nfrab1 2691	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝜑}
3	2	nfcri 2346	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}
4	3	nfex 1663	. . 3 ⊢ Ⅎ𝑥∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}
5		rabid 2687	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑥 ∈ On ∧ 𝜑))
6		elex2 2796	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
7	5, 6	sylbir 135	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝜑) → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
8	7	ex 115	. . 3 ⊢ (𝑥 ∈ On → (𝜑 → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}))
9	4, 8	rexlimi 2621	. 2 ⊢ (∃𝑥 ∈ On 𝜑 → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
10		onintonm 4586	. 2 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}) → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On)
11	1, 9, 10	sylancr 414	1 ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On)

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1518   ∈ wcel 2180  ∃wrex 2489  {crab 2492   ⊆ wss 3177  ∩ cint 3902  Oncon0 4431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-rab 2497  df-v 2781  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-uni 3868  df-int 3903  df-tr 4162  df-iord 4434  df-on 4436  df-suc 4439

This theorem is referenced by:  cardcl  7321