Theorem dfon2lem2 35785

Description: Lemma for dfon2 35793. (Contributed by Scott Fenton, 28-Feb-2011.)

Assertion

Ref	Expression
dfon2lem2	⊢ ∪ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝐴

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem dfon2lem2

Step	Hyp	Ref	Expression
1		simp1 1137	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓) → 𝑥 ⊆ 𝐴)
2	1	ss2abi 4067	. . 3 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴}
3		df-pw 4602	. . 3 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
4	2, 3	sseqtrri 4033	. 2 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝒫 𝐴
5		sspwuni 5100	. 2 ⊢ ({𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝒫 𝐴 ↔ ∪ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝐴)
6	4, 5	mpbi 230	1 ⊢ ∪ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝐴

Syntax hints:   ∧ w3a 1087  {cab 2714   ⊆ wss 3951  𝒫 cpw 4600  ∪ cuni 4907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-v 3482  df-ss 3968  df-pw 4602  df-uni 4908

This theorem is referenced by:  dfon2lem3  35786  dfon2lem7  35790