Theorem abssexg 3934

Description: Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
abssexg	⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem abssexg

Step	Hyp	Ref	Expression
1		pwexg 3933	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
2		df-pw 3361	. . . 4 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
3	2	eleq1i 2103	. . 3 ⊢ (𝒫 𝐴 ∈ V ↔ {𝑥 ∣ 𝑥 ⊆ 𝐴} ∈ V)
4		simpl 102	. . . . 5 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝜑) → 𝑥 ⊆ 𝐴)
5	4	ss2abi 3012	. . . 4 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴}
6		ssexg 3896	. . . 4 ⊢ (({𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴} ∧ {𝑥 ∣ 𝑥 ⊆ 𝐴} ∈ V) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V)
7	5, 6	mpan 400	. . 3 ⊢ ({𝑥 ∣ 𝑥 ⊆ 𝐴} ∈ V → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V)
8	3, 7	sylbi 114	. 2 ⊢ (𝒫 𝐴 ∈ V → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V)
9	1, 8	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1393  {cab 2026  Vcvv 2557   ⊆ wss 2917  𝒫 cpw 3359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361

This theorem is referenced by: (None)