Theorem ssn0rex 4309

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 4004	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 𝑥 ∈ 𝐴))
2		n0rex 4308	. 2 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴)
3	1, 2	impel 515	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐵 𝑥 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145   ≠ wne 2957  ∃wrex 3088   ⊆ wss 3902  ∅c0 4282

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-ne 2958  df-rex 3089  df-dif 3905  df-ss 3919  df-nul 4283

This theorem is used by:  uhgrvd00  29980