Theorem ssel 3268

Description: Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
ssel	⊢ (A ⊆ B → (C ∈ A → C ∈ B))

Proof of Theorem ssel

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 3263	. . . . . 6 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))
2	1	biimpi 186	. . . . 5 ⊢ (A ⊆ B → ∀x(x ∈ A → x ∈ B))
3	2	19.21bi 1758	. . . 4 ⊢ (A ⊆ B → (x ∈ A → x ∈ B))
4	3	anim2d 548	. . 3 ⊢ (A ⊆ B → ((x = C ∧ x ∈ A) → (x = C ∧ x ∈ B)))
5	4	eximdv 1622	. 2 ⊢ (A ⊆ B → (∃x(x = C ∧ x ∈ A) → ∃x(x = C ∧ x ∈ B)))
6		df-clel 2349	. 2 ⊢ (C ∈ A ↔ ∃x(x = C ∧ x ∈ A))
7		df-clel 2349	. 2 ⊢ (C ∈ B ↔ ∃x(x = C ∧ x ∈ B))
8	5, 6, 7	3imtr4g 261	1 ⊢ (A ⊆ B → (C ∈ A → C ∈ B))

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ⊆ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  ssel2  3269  sseli  3270  sseld  3273  sstr2  3280  ssralv  3331  ssrexv  3332  ralss  3333  rexss  3334  ssconb  3400  sscon  3401  ssdif  3402  unss1  3433  ssrin  3481  difin2  3517  reuss2  3536  reupick  3540  sssn  3865  uniss  3913  ss2iun  3985  ssiun  4009  iinss  4018  ssofss  4077  unipw  4118  sspwb  4119  pwadjoin  4120  eqpw1uni  4331  ssopab2b  4714  ssrel  4845  xpss12  4856  cnvss  4886  dmss  4907  dmcosseq  4974  ssreseq  4998  iss  5001  resopab2  5002  ssrnres  5060  dfco2a  5082  cores  5085  funssres  5145  fununi  5161  funimass3  5405  funfvima2  5461  f1elima  5475  resoprab2  5583  clos1conn  5880