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Theorem dfon2lem2 35785
Description: Lemma for dfon2 35793. (Contributed by Scott Fenton, 28-Feb-2011.)
Assertion
Ref Expression
dfon2lem2 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝐴
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem dfon2lem2
StepHypRef Expression
1 simp1 1137 . . . 4 ((𝑥𝐴𝜑𝜓) → 𝑥𝐴)
21ss2abi 4067 . . 3 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ {𝑥𝑥𝐴}
3 df-pw 4602 . . 3 𝒫 𝐴 = {𝑥𝑥𝐴}
42, 3sseqtrri 4033 . 2 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝒫 𝐴
5 sspwuni 5100 . 2 ({𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝒫 𝐴 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝐴)
64, 5mpbi 230 1 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝐴
Colors of variables: wff setvar class
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
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