Theorem ssn0rex 4355

Description: There is an element in a class with a nonempty subclass which is an element of the subclass. (Contributed by AV, 17-Dec-2020.)

Assertion

Ref	Expression
ssn0rex	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐵 𝑥 ∈ 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ssn0rex

Step	Hyp	Ref	Expression
1		ssrexv 4051	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 𝑥 ∈ 𝐴))
2		n0rex 4354	. 2 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴)
3	1, 2	impel 506	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐵 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106   ≠ wne 2940  ∃wrex 3070   ⊆ wss 3948  ∅c0 4322

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-rex 3071  df-v 3476  df-dif 3951  df-in 3955  df-ss 3965  df-nul 4323

This theorem is referenced by:  uhgrvd00  28788