Theorem pssexg 42221

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 4064	. 2 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵)
2		ssexg 5281	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
3	1, 2	sylan 580	1 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  Vcvv 3450   ⊆ wss 3917   ⊊ wpss 3918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-in 3924  df-ss 3934  df-pss 3937

This theorem is referenced by: (None)