Theorem rabsnel 30270

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 4561	. . 3 ⊢ 𝐵 ∈ {𝐵}
3		eleq2 2878	. . 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 3623	. 2 ⊢ (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝐵 ∈ 𝐴)
6	4, 5	syl 17	1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3441  {csn 4525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-sn 4526

This theorem is referenced by:  ddemeas  31605