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Theorem brssr 37359
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 37358 . . . . 5 Rel S
21brrelex1i 5730 . . . 4 (𝐴 S 𝐵𝐴 ∈ V)
32adantl 482 . . 3 ((𝐵𝑉𝐴 S 𝐵) → 𝐴 ∈ V)
4 simpl 483 . . 3 ((𝐵𝑉𝐴 S 𝐵) → 𝐵𝑉)
53, 4jca 512 . 2 ((𝐵𝑉𝐴 S 𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
6 ssexg 5322 . . . 4 ((𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
7 simpr 485 . . . 4 ((𝐴𝐵𝐵𝑉) → 𝐵𝑉)
86, 7jca 512 . . 3 ((𝐴𝐵𝐵𝑉) → (𝐴 ∈ V ∧ 𝐵𝑉))
98ancoms 459 . 2 ((𝐵𝑉𝐴𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
10 sseq1 4006 . . 3 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
11 sseq2 4007 . . 3 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
12 df-ssr 37356 . . 3 S = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1310, 11, 12brabg 5538 . 2 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 S 𝐵𝐴𝐵))
145, 9, 13pm5.21nd 800 1 (𝐵𝑉 → (𝐴 S 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  Vcvv 3474  wss 3947   class class class wbr 5147   S cssr 37034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-ssr 37356
This theorem is referenced by:  brssrid  37360  brssrres  37362  brcnvssr  37364  extssr  37367  dfrefrels2  37371  dfsymrels2  37403  dftrrels2  37433
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