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Theorem extssr 37017
Description: Property of subset relation, see also extid 36817, extep 36789 and the comment of df-ssr 37006. (Contributed by Peter Mazsa, 10-Jul-2019.)
Assertion
Ref Expression
extssr ((𝐴𝑉𝐵𝑊) → ([𝐴] S = [𝐵] S ↔ 𝐴 = 𝐵))

Proof of Theorem extssr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 brssr 37009 . . . 4 (𝐴𝑉 → (𝑥 S 𝐴𝑥𝐴))
2 brssr 37009 . . . 4 (𝐵𝑊 → (𝑥 S 𝐵𝑥𝐵))
31, 2bi2bian9 640 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝑥 S 𝐴𝑥 S 𝐵) ↔ (𝑥𝐴𝑥𝐵)))
43albidv 1924 . 2 ((𝐴𝑉𝐵𝑊) → (∀𝑥(𝑥 S 𝐴𝑥 S 𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵)))
5 relssr 37008 . . 3 Rel S
6 releccnveq 36764 . . 3 ((Rel S ∧ Rel S ) → ([𝐴] S = [𝐵] S ↔ ∀𝑥(𝑥 S 𝐴𝑥 S 𝐵)))
75, 5, 6mp2an 691 . 2 ([𝐴] S = [𝐵] S ↔ ∀𝑥(𝑥 S 𝐴𝑥 S 𝐵))
8 ssext 5412 . 2 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
94, 7, 83bitr4g 314 1 ((𝐴𝑉𝐵𝑊) → ([𝐴] S = [𝐵] S ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wcel 2107  wss 3911   class class class wbr 5106  ccnv 5633  Rel wrel 5639  [cec 8649   S cssr 36683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ec 8653  df-ssr 37006
This theorem is referenced by: (None)
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