Theorem rabsnel 32173

Description: Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by Thierry Arnoux, 15-Sep-2018.)

Hypothesis

Ref	Expression
rabsnel.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
rabsnel	⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝐵 ∈ 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabsnel

Step	Hyp	Ref	Expression
1		rabsnel.1	. . . 4 ⊢ 𝐵 ∈ V
2	1	snid 4664	. . 3 ⊢ 𝐵 ∈ {𝐵}
3		eleq2 2821	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝐵 ∈ {𝐵}))
4	2, 3	mpbiri 258	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
5		elrabi 3677	. 2 ⊢ (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝐵 ∈ 𝐴)
6	4, 5	syl 17	1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  {crab 3431  Vcvv 3473  {csn 4628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-sn 4629

This theorem is referenced by:  ddemeas  33698