Theorem trss 5173

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 5168	. 2 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴)
2		sseq1 3991	. . 3 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
3	2	rspccv 3619	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
4	1, 3	sylbi 219	1 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2110  ∀wral 3138   ⊆ wss 3935  Tr wtr 5164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3496  df-in 3942  df-ss 3951  df-uni 4832  df-tr 5165

This theorem is referenced by:  trin  5174  triun  5177  triin  5179  trintss  5181  tz7.2  5533  ordelss  6201  ordelord  6207  tz7.7  6211  trsucss  6270  omsinds  7594  tc2  9178  tcel  9181  r1ord3g  9202  r1ord2  9204  r1pwss  9207  rankwflemb  9216  r1elwf  9219  r1elssi  9228  uniwf  9242  itunitc1  9836  wunelss  10124  tskr1om2  10184  tskuni  10199  tskurn  10205  gruelss  10210  dfon2lem6  33028  dfon2lem9  33031  setindtr  39614  dford3lem1  39616  ordelordALT  40864  trsspwALT  41145  trsspwALT2  41146  trsspwALT3  41147  pwtrVD  41151  ordelordALTVD  41194