Theorem rabsssn 4626

Description: Conditions for a restricted class abstraction to be a subset of a singleton, i.e. to be a singleton or the empty set. (Contributed by AV, 18-Apr-2019.)

Assertion

Ref	Expression
rabsssn	⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ {𝑋} ↔ ∀𝑥 ∈ 𝑉 (𝜑 → 𝑥 = 𝑋))

Distinct variable group:   𝑥,𝑋

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem rabsssn

Step	Hyp	Ref	Expression
1		df-rab 3401	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)}
2		df-sn 4582	. . 3 ⊢ {𝑋} = {𝑥 ∣ 𝑥 = 𝑋}
3	1, 2	sseq12i 3965	. 2 ⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ {𝑋} ↔ {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝑥 = 𝑋})
4		ss2ab 4014	. 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝑥 = 𝑋} ↔ ∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) → 𝑥 = 𝑋))
5		impexp 450	. . . 4 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝜑) → 𝑥 = 𝑋) ↔ (𝑥 ∈ 𝑉 → (𝜑 → 𝑥 = 𝑋)))
6	5	albii 1821	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) → 𝑥 = 𝑋) ↔ ∀𝑥(𝑥 ∈ 𝑉 → (𝜑 → 𝑥 = 𝑋)))
7		df-ral 3053	. . 3 ⊢ (∀𝑥 ∈ 𝑉 (𝜑 → 𝑥 = 𝑋) ↔ ∀𝑥(𝑥 ∈ 𝑉 → (𝜑 → 𝑥 = 𝑋)))
8	6, 7	bitr4i 278	. 2 ⊢ (∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) → 𝑥 = 𝑋) ↔ ∀𝑥 ∈ 𝑉 (𝜑 → 𝑥 = 𝑋))
9	3, 4, 8	3bitri 297	1 ⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ {𝑋} ↔ ∀𝑥 ∈ 𝑉 (𝜑 → 𝑥 = 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  {crab 3400   ⊆ wss 3902  {csn 4581

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rab 3401  df-ss 3919  df-sn 4582

This theorem is referenced by:  constrfin  33905  suppmptcfin  48689  linc1  48738