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Theorem extssr 35194
Description: Property of subset relation, see also extid 35012, extep 34984 and the comment of df-ssr 35183. (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 35186 . . . 4 (𝐴𝑉 → (𝑥 S 𝐴𝑥𝐴))
2 brssr 35186 . . . 4 (𝐵𝑊 → (𝑥 S 𝐵𝑥𝐵))
31, 2bi2bian9 628 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝑥 S 𝐴𝑥 S 𝐵) ↔ (𝑥𝐴𝑥𝐵)))
43albidv 1879 . 2 ((𝐴𝑉𝐵𝑊) → (∀𝑥(𝑥 S 𝐴𝑥 S 𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵)))
5 relssr 35185 . . 3 Rel S
6 releccnveq 34963 . . 3 ((Rel S ∧ Rel S ) → ([𝐴] S = [𝐵] S ↔ ∀𝑥(𝑥 S 𝐴𝑥 S 𝐵)))
75, 5, 6mp2an 679 . 2 ([𝐴] S = [𝐵] S ↔ ∀𝑥(𝑥 S 𝐴𝑥 S 𝐵))
8 ssext 5198 . 2 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
94, 7, 83bitr4g 306 1 ((𝐴𝑉𝐵𝑊) → ([𝐴] S = [𝐵] S ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wal 1505   = wceq 1507  wcel 2050  wss 3823   class class class wbr 4923  ccnv 5400  Rel wrel 5406  [cec 8081   S cssr 34900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-br 4924  df-opab 4986  df-xp 5407  df-rel 5408  df-cnv 5409  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-ec 8085  df-ssr 35183
This theorem is referenced by: (None)
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