Theorem pssexg 39187

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 4069	. 2 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵)
2		ssexg 5224	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
3	1, 2	sylan 582	1 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2113  Vcvv 3493   ⊆ wss 3933   ⊊ wpss 3934

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5200

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-rab 3146  df-v 3495  df-in 3940  df-ss 3949  df-pss 3951

This theorem is referenced by: (None)