Theorem sseldd 3001

Description: Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)

Hypotheses

Ref	Expression
sseld.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
sseldd.2	⊢ (𝜑 → 𝐶 ∈ 𝐴)

Assertion

Ref	Expression
sseldd	⊢ (𝜑 → 𝐶 ∈ 𝐵)

Proof of Theorem sseldd

Step	Hyp	Ref	Expression
1		sseldd.2	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
2		sseld.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3	2	sseld 2999	. 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))
4	1, 3	mpd 13	1 ⊢ (𝜑 → 𝐶 ∈ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 1434   ⊆ wss 2974

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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987

This theorem is referenced by:  frirrg  4113  ordtri2or2exmid  4322  riotass  5526  tfrcldm  6012  eroveu  6263  eroprf  6265  findcard2d  6425  findcard2sd  6426  undiffi  6443  suplub2ti  6473  nnppipi  6595  archnqq  6669  prarloclemlt  6745  suprubex  8096  suprzclex  8526  fzssp1  9161  elfzoelz  9234  fzofzp1  9313  fzostep1  9323  frecuzrdgg  9498  frecuzrdgdomlem  9499  frecuzrdgsuctlem  9505  iseqvalt  9532  isermono  9553  bcm1k  9784  fimaxre2  10247  zssinfcl  10488