Theorem pssexg 42561

Description: The proper subset of a set is also a set. (Contributed by Steven Nguyen, 17-Jul-2022.)

Assertion

Ref	Expression
pssexg	⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)

Proof of Theorem pssexg

Step	Hyp	Ref	Expression
1		pssss 4051	. 2 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵)
2		ssexg 5269	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
3	1, 2	sylan 581	1 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  Vcvv 3441   ⊆ wss 3902   ⊊ wpss 3903

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-ext 2709  ax-sep 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3401  df-v 3443  df-in 3909  df-ss 3919  df-pss 3922

This theorem is referenced by: (None)