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Theorem bj-rabtrAUTO 36302
Description: Proof of bj-rabtr 36300 found automatically by the Metamath program "MM-PA> IMPROVE ALL / DEPTH 3 / 3" command followed by "MM-PA> MINIMIZE_WITH *". (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-rabtrAUTO {𝑥𝐴 ∣ ⊤} = 𝐴
Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-rabtrAUTO
StepHypRef Expression
1 ssrab2 4069 . 2 {𝑥𝐴 ∣ ⊤} ⊆ 𝐴
2 ssid 3996 . . . . 5 𝐴𝐴
32a1i 11 . . . 4 (⊤ → 𝐴𝐴)
4 simpl 482 . . . 4 ((⊤ ∧ 𝑥𝐴) → ⊤)
53, 4ssrabdv 4063 . . 3 (⊤ → 𝐴 ⊆ {𝑥𝐴 ∣ ⊤})
65mptru 1540 . 2 𝐴 ⊆ {𝑥𝐴 ∣ ⊤}
71, 6eqssi 3990 1 {𝑥𝐴 ∣ ⊤} = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wtru 1534  wcel 2098  {crab 3424  wss 3940
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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rab 3425  df-v 3468  df-in 3947  df-ss 3957
This theorem is referenced by: (None)
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