Theorem bj-rabtrALTALT 32120

Description: Alternate proof of bj-rabtr 32118. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-rabtrALTALT	⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-rabtrALTALT

Step	Hyp	Ref	Expression
1		ssrab2 3650	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} ⊆ 𝐴
2		ssid 3587	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		tru 1479	. . . 4 ⊢ ⊤
4	3	rgenw 2908	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ⊤
5		ssrab 3643	. . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ⊤))
6	2, 4, 5	mpbir2an 957	. 2 ⊢ 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤}
7	1, 6	eqssi 3584	1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

Syntax hints:   = wceq 1475  ⊤wtru 1476  ∀wral 2896  {crab 2900   ⊆ wss 3540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-in 3547  df-ss 3554

This theorem is referenced by: (None)