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Theorem dfon2lem2 31813
Description: Lemma for dfon2 31821. (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 1081 . . . 4 ((𝑥𝐴𝜑𝜓) → 𝑥𝐴)
21ss2abi 3707 . . 3 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ {𝑥𝑥𝐴}
3 df-pw 4193 . . 3 𝒫 𝐴 = {𝑥𝑥𝐴}
42, 3sseqtr4i 3671 . 2 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝒫 𝐴
5 sspwuni 4643 . 2 ({𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝒫 𝐴 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝐴)
64, 5mpbi 220 1 {𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  w3a 1054  {cab 2637  wss 3607  𝒫 cpw 4191   cuni 4468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-in 3614  df-ss 3621  df-pw 4193  df-uni 4469
This theorem is referenced by:  dfon2lem3  31814  dfon2lem7  31818
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