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Theorem dfon2lem2 33033
 Description: Lemma for dfon2 33041. (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 1132 . . . 4 ((𝑥𝐴𝜑𝜓) → 𝑥𝐴)
21ss2abi 4018 . . 3 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ {𝑥𝑥𝐴}
3 df-pw 4513 . . 3 𝒫 𝐴 = {𝑥𝑥𝐴}
42, 3sseqtrri 3979 . 2 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝒫 𝐴
5 sspwuni 4994 . 2 ({𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝒫 𝐴 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝐴)
64, 5mpbi 232 1 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∧ w3a 1083  {cab 2798   ⊆ wss 3909  𝒫 cpw 4511  ∪ cuni 4810 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-v 3472  df-in 3916  df-ss 3926  df-pw 4513  df-uni 4811 This theorem is referenced by:  dfon2lem3  33034  dfon2lem7  33038
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