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Theorem brssr 38482
Description: The subset relation and subclass relationship (df-ss 3979) 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 38481 . . . . 5 Rel S
21brrelex1i 5744 . . . 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 5328 . . . 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 4020 . . 3 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
11 sseq2 4021 . . 3 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
12 df-ssr 38479 . . 3 S = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1310, 11, 12brabg 5548 . 2 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 S 𝐵𝐴𝐵))
145, 9, 13pm5.21nd 802 1 (𝐵𝑉 → (𝐴 S 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2105  Vcvv 3477  wss 3962   class class class wbr 5147   S cssr 38164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-ssr 38479
This theorem is referenced by:  brssrid  38483  brssrres  38485  brcnvssr  38487  extssr  38490  dfrefrels2  38494  dfsymrels2  38526  dftrrels2  38556
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