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Theorem brssr 34873
Description: The subset relation and subclass relationship (df-ss 3805) are the same, that is, (𝐴 S 𝐵𝐴𝐵) when 𝐵 is a set. (Contributed by Peter Mazsa, 31-Jul-2019.)
Assertion
Ref Expression
brssr (𝐵𝑉 → (𝐴 S 𝐵𝐴𝐵))

Proof of Theorem brssr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relssr 34872 . . . . 5 Rel S
21brrelex1i 5406 . . . 4 (𝐴 S 𝐵𝐴 ∈ V)
32adantl 475 . . 3 ((𝐵𝑉𝐴 S 𝐵) → 𝐴 ∈ V)
4 simpl 476 . . 3 ((𝐵𝑉𝐴 S 𝐵) → 𝐵𝑉)
53, 4jca 507 . 2 ((𝐵𝑉𝐴 S 𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
6 ssexg 5041 . . . 4 ((𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
7 simpr 479 . . . 4 ((𝐴𝐵𝐵𝑉) → 𝐵𝑉)
86, 7jca 507 . . 3 ((𝐴𝐵𝐵𝑉) → (𝐴 ∈ V ∧ 𝐵𝑉))
98ancoms 452 . 2 ((𝐵𝑉𝐴𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
10 sseq1 3844 . . 3 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
11 sseq2 3845 . . 3 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
12 df-ssr 34870 . . 3 S = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1310, 11, 12brabg 5231 . 2 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 S 𝐵𝐴𝐵))
145, 9, 13pm5.21nd 792 1 (𝐵𝑉 → (𝐴 S 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wcel 2106  Vcvv 3397  wss 3791   class class class wbr 4886   S cssr 34603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4887  df-opab 4949  df-xp 5361  df-rel 5362  df-ssr 34870
This theorem is referenced by:  brssrid  34874  brssrres  34876  brcnvssr  34878  extssr  34881  dfrefrels2  34885  dfsymrels2  34913  dftrrels2  34943
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