Theorem abssexg 4168

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 4166	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
2		df-pw 3568	. . . 4 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
3	2	eleq1i 2236	. . 3 ⊢ (𝒫 𝐴 ∈ V ↔ {𝑥 ∣ 𝑥 ⊆ 𝐴} ∈ V)
4		simpl 108	. . . . 5 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝜑) → 𝑥 ⊆ 𝐴)
5	4	ss2abi 3219	. . . 4 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴}
6		ssexg 4128	. . . 4 ⊢ (({𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴} ∧ {𝑥 ∣ 𝑥 ⊆ 𝐴} ∈ V) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V)
7	5, 6	mpan 422	. . 3 ⊢ ({𝑥 ∣ 𝑥 ⊆ 𝐴} ∈ V → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V)
8	3, 7	sylbi 120	. 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 103   ∈ wcel 2141  {cab 2156  Vcvv 2730   ⊆ wss 3121  𝒫 cpw 3566

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568

This theorem is referenced by:  pmex  6631  tgval  12843