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Theorem extssr 38510
Description: Property of subset relation, see also extid 38311, extep 38284 and the comment of df-ssr 38499. (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 38502 . . . 4 (𝐴𝑉 → (𝑥 S 𝐴𝑥𝐴))
2 brssr 38502 . . . 4 (𝐵𝑊 → (𝑥 S 𝐵𝑥𝐵))
31, 2bi2bian9 640 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝑥 S 𝐴𝑥 S 𝐵) ↔ (𝑥𝐴𝑥𝐵)))
43albidv 1920 . 2 ((𝐴𝑉𝐵𝑊) → (∀𝑥(𝑥 S 𝐴𝑥 S 𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵)))
5 relssr 38501 . . 3 Rel S
6 releccnveq 38259 . . 3 ((Rel S ∧ Rel S ) → ([𝐴] S = [𝐵] S ↔ ∀𝑥(𝑥 S 𝐴𝑥 S 𝐵)))
75, 5, 6mp2an 692 . 2 ([𝐴] S = [𝐵] S ↔ ∀𝑥(𝑥 S 𝐴𝑥 S 𝐵))
8 ssext 5459 . 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 206  wa 395  wal 1538   = wceq 1540  wcel 2108  wss 3951   class class class wbr 5143  ccnv 5684  Rel wrel 5690  [cec 8743   S cssr 38185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-ssr 38499
This theorem is referenced by: (None)
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