Theorem ssel2 3020

Description: Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)

Assertion

Ref	Expression
ssel2	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐵)

Proof of Theorem ssel2

Step	Hyp	Ref	Expression
1		ssel 3019	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))
2	1	imp 122	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 102   ∈ wcel 1438   ⊆ wss 2999

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-ss 3012

This theorem is referenced by:  elnn  4418  funimass4  5349  fvelimab  5354  ssimaex  5359  funconstss  5411  rexima  5526  ralima  5527  1st2nd  5943  f1o2ndf1  5985  tfri1dALT  6108  eldju1st  6752  lbinf  8399  dfinfre  8407  lbzbi  9091  elfzom1elp1fzo  9601  ssfzo12  9623  seq3split  9895  iseqsplit  9896  shftlem  10238