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Theorem brssr 37973
Description: The subset relation and subclass relationship (df-ss 3964) 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 37972 . . . . 5 Rel S
21brrelex1i 5734 . . . 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 5323 . . . 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 4005 . . 3 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
11 sseq2 4006 . . 3 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
12 df-ssr 37970 . . 3 S = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1310, 11, 12brabg 5541 . 2 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 S 𝐵𝐴𝐵))
145, 9, 13pm5.21nd 801 1 (𝐵𝑉 → (𝐴 S 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wcel 2099  Vcvv 3471  wss 3947   class class class wbr 5148   S cssr 37651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-ssr 37970
This theorem is referenced by:  brssrid  37974  brssrres  37976  brcnvssr  37978  extssr  37981  dfrefrels2  37985  dfsymrels2  38017  dftrrels2  38047
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