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Theorem extssr 37184
Description: Property of subset relation, see also extid 36984, extep 36956 and the comment of df-ssr 37173. (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 37176 . . . 4 (𝐴𝑉 → (𝑥 S 𝐴𝑥𝐴))
2 brssr 37176 . . . 4 (𝐵𝑊 → (𝑥 S 𝐵𝑥𝐵))
31, 2bi2bian9 639 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝑥 S 𝐴𝑥 S 𝐵) ↔ (𝑥𝐴𝑥𝐵)))
43albidv 1923 . 2 ((𝐴𝑉𝐵𝑊) → (∀𝑥(𝑥 S 𝐴𝑥 S 𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵)))
5 relssr 37175 . . 3 Rel S
6 releccnveq 36931 . . 3 ((Rel S ∧ Rel S ) → ([𝐴] S = [𝐵] S ↔ ∀𝑥(𝑥 S 𝐴𝑥 S 𝐵)))
75, 5, 6mp2an 690 . 2 ([𝐴] S = [𝐵] S ↔ ∀𝑥(𝑥 S 𝐴𝑥 S 𝐵))
8 ssext 5447 . 2 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
94, 7, 83bitr4g 313 1 ((𝐴𝑉𝐵𝑊) → ([𝐴] S = [𝐵] S ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wcel 2106  wss 3944   class class class wbr 5141  ccnv 5668  Rel wrel 5674  [cec 8684   S cssr 36851
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-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
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-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8688  df-ssr 37173
This theorem is referenced by: (None)
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