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Theorem dfon2lem2 35289
Description: Lemma for dfon2 35297. (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 1133 . . . 4 ((𝑥𝐴𝜑𝜓) → 𝑥𝐴)
21ss2abi 4058 . . 3 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ {𝑥𝑥𝐴}
3 df-pw 4599 . . 3 𝒫 𝐴 = {𝑥𝑥𝐴}
42, 3sseqtrri 4014 . 2 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝒫 𝐴
5 sspwuni 5096 . 2 ({𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝒫 𝐴 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝐴)
64, 5mpbi 229 1 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  w3a 1084  {cab 2703  wss 3943  𝒫 cpw 4597   cuni 4902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-v 3470  df-in 3950  df-ss 3960  df-pw 4599  df-uni 4903
This theorem is referenced by:  dfon2lem3  35290  dfon2lem7  35294
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