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Theorem bj-rabtrAUTO 36915
Description: Proof of bj-rabtr 36913 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 4090 . 2 {𝑥𝐴 ∣ ⊤} ⊆ 𝐴
2 ssid 4018 . . . . 5 𝐴𝐴
32a1i 11 . . . 4 (⊤ → 𝐴𝐴)
4 simpl 482 . . . 4 ((⊤ ∧ 𝑥𝐴) → ⊤)
53, 4ssrabdv 4084 . . 3 (⊤ → 𝐴 ⊆ {𝑥𝐴 ∣ ⊤})
65mptru 1544 . 2 𝐴 ⊆ {𝑥𝐴 ∣ ⊤}
71, 6eqssi 4012 1 {𝑥𝐴 ∣ ⊤} = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wtru 1538  wcel 2106  {crab 3433  wss 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rab 3434  df-ss 3980
This theorem is referenced by: (None)
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