Theorem dfss3 2929

Description: Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)

Assertion

Ref	Expression
dfss3	⊢ (A ⊆ B ↔ ∀x ∈ A x ∈ B)

Distinct variable groups:   x,A   x,B

Proof of Theorem dfss3

Step	Hyp	Ref	Expression
1		dfss2 2928	. 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))
2		df-ral 2305	. 2 ⊢ (∀x ∈ A x ∈ B ↔ ∀x(x ∈ A → x ∈ B))
3	1, 2	bitr4i 176	1 ⊢ (A ⊆ B ↔ ∀x ∈ A x ∈ B)

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   ∈ wcel 1390  ∀wral 2300   ⊆ wss 2911

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-in 2918  df-ss 2925

This theorem is referenced by:  ssrab  3012  eqsnm  3517  uni0b  3596  uni0c  3597  ssint  3622  ssiinf  3697  sspwuni  3730  dftr3  3849  tfis  4249  rninxp  4707  fnres  4958  eqfnfv3  5210  funimass3  5226  ffvresb  5271  tfrlemibxssdm  5882  bdss  9253