Theorem trss 5221

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 5216	. 2 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴)
2		sseq1 3933	. . 3 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
3	2	rspccv 3573	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
4	1, 3	sylbi 220	1 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895  Tr wtr 5211

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-ss 3896  df-uni 4867  df-tr 5212

This theorem is used by:  trun  5222  trin  5223  triun  5226  triin  5228  trintss  5230  tz7.2  5633  trpred  6334  ordelss  6378  ordelord  6384  tz7.7  6388  trsucss  6453  tc2  9745  tcel  9748  r1ord3g  9790  r1ord2  9792  r1pwss  9795  rankwflemb  9804  rankwflembOLD  9805  r1elwf  9808  r1elssi  9817  uniwf  9832  itunitc1  10502  wunelss  10797  tskhf  10857  tskuni  10872  tskurn  10878  gruelss  10883  tz9.1regs  35859  dfon2lem6  36643  dfon2lem9  36646  nmuladdss  37055  axtco2g  37358  tr0elw  37365  tr0el  37366  ttctr2  37375  ttciunun  37392  setindtr  44125  dford3lem1  44127  ordelordALT  45614  trsspwALT  45894  trsspwALT2  45895  trsspwALT3  45896  pwtrVD  45900  ordelordALTVD  45943  ralabso  46052  rexabso  46053  modelaxrep  46065  omelaxinf2  46073