Theorem rabsnel 33150

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 4622	. . 3 ⊢ 𝐵 ∈ {𝐵}
3		eleq2 2852	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝐵 ∈ {𝐵}))
4	2, 3	mpbiri 261	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
5		elrabi 3643	. 2 ⊢ (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝐵 ∈ 𝐴)
6	4, 5	syl 18	1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝐵 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146  {crab 3415  Vcvv 3453  {csn 4587

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3455  df-sn 4588

This theorem is used by:  ddemeas  34922