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Theorem dfon2lem2 31992
Description: Lemma for dfon2 32000. (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 1131 . . . 4 ((𝑥𝐴𝜑𝜓) → 𝑥𝐴)
21ss2abi 3813 . . 3 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ {𝑥𝑥𝐴}
3 df-pw 4302 . . 3 𝒫 𝐴 = {𝑥𝑥𝐴}
42, 3sseqtr4i 3777 . 2 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝒫 𝐴
5 sspwuni 4761 . 2 ({𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝒫 𝐴 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝐴)
64, 5mpbi 220 1 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  w3a 1072  {cab 2744  wss 3713  𝒫 cpw 4300   cuni 4586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-v 3340  df-in 3720  df-ss 3727  df-pw 4302  df-uni 4587
This theorem is referenced by:  dfon2lem3  31993  dfon2lem7  31997
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