Theorem trss 4794

Description: An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.) (Proof shortened by JJ, 26-Jul-2021.)

Assertion

Ref	Expression
trss	⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))

Proof of Theorem trss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dftr3 4789	. 2 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴)
2		sseq1 3659	. . 3 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
3	2	rspccv 3337	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
4	1, 3	sylbi 207	1 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2030  ∀wral 2941   ⊆ wss 3607  Tr wtr 4785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-in 3614  df-ss 3621  df-uni 4469  df-tr 4786

This theorem is referenced by:  trin  4796  triun  4799  trintss  4802  tz7.2  5127  ordelss  5777  ordelord  5783  tz7.7  5787  trsucss  5849  omsinds  7126  tc2  8656  tcel  8659  r1ord3g  8680  r1ord2  8682  r1pwss  8685  rankwflemb  8694  r1elwf  8697  r1elssi  8706  uniwf  8720  itunitc1  9280  wunelss  9568  tskr1om2  9628  tskuni  9643  tskurn  9649  gruelss  9654  dfon2lem6  31817  dfon2lem9  31820  setindtr  37908  dford3lem1  37910  ordelordALT  39064  trsspwALT  39362  trsspwALT2  39363  trsspwALT3  39364  pwtrVD  39373  ordelordALTVD  39417