Theorem pssexg 43281

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 4025	. 2 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵)
2		ssexg 5280	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
3	1, 2	sylan 592	1 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895   ⊊ wpss 4013

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 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-in 4091

This theorem is used by: (None)