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Theorem brssr 37874
Description: The subset relation and subclass relationship (df-ss 3958) 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 37873 . . . . 5 Rel S
21brrelex1i 5723 . . . 4 (𝐴 S 𝐵𝐴 ∈ V)
32adantl 481 . . 3 ((𝐵𝑉𝐴 S 𝐵) → 𝐴 ∈ V)
4 simpl 482 . . 3 ((𝐵𝑉𝐴 S 𝐵) → 𝐵𝑉)
53, 4jca 511 . 2 ((𝐵𝑉𝐴 S 𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
6 ssexg 5314 . . . 4 ((𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
7 simpr 484 . . . 4 ((𝐴𝐵𝐵𝑉) → 𝐵𝑉)
86, 7jca 511 . . 3 ((𝐴𝐵𝐵𝑉) → (𝐴 ∈ V ∧ 𝐵𝑉))
98ancoms 458 . 2 ((𝐵𝑉𝐴𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
10 sseq1 4000 . . 3 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
11 sseq2 4001 . . 3 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
12 df-ssr 37871 . . 3 S = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1310, 11, 12brabg 5530 . 2 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 S 𝐵𝐴𝐵))
145, 9, 13pm5.21nd 799 1 (𝐵𝑉 → (𝐴 S 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wcel 2098  Vcvv 3466  wss 3941   class class class wbr 5139   S cssr 37549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-ssr 37871
This theorem is referenced by:  brssrid  37875  brssrres  37877  brcnvssr  37879  extssr  37882  dfrefrels2  37886  dfsymrels2  37918  dftrrels2  37948
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