Theorem trss 5270

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 5265	. 2 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴)
2		sseq1 4009	. . 3 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
3	2	rspccv 3619	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
4	1, 3	sylbi 217	1 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2108  ∀wral 3061   ⊆ wss 3951  Tr wtr 5259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-v 3482  df-ss 3968  df-uni 4908  df-tr 5260

This theorem is referenced by:  trin  5271  triun  5274  triin  5276  trintss  5278  tz7.2  5668  trpred  6352  ordelss  6400  ordelord  6406  tz7.7  6410  trsucss  6472  tc2  9782  tcel  9785  r1ord3g  9819  r1ord2  9821  r1pwss  9824  rankwflemb  9833  r1elwf  9836  r1elssi  9845  uniwf  9859  itunitc1  10460  wunelss  10748  tskr1om2  10808  tskuni  10823  tskurn  10829  gruelss  10834  dfon2lem6  35789  dfon2lem9  35792  setindtr  43036  dford3lem1  43038  ordelordALT  44557  trsspwALT  44838  trsspwALT2  44839  trsspwALT3  44840  pwtrVD  44844  ordelordALTVD  44887  ralabso  44985  rexabso  44986  modelaxrep  44998  omelaxinf2  45006