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Theorem extssr 35854
Description: Property of subset relation, see also extid 35673, extep 35645 and the comment of df-ssr 35843. (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 35846 . . . 4 (𝐴𝑉 → (𝑥 S 𝐴𝑥𝐴))
2 brssr 35846 . . . 4 (𝐵𝑊 → (𝑥 S 𝐵𝑥𝐵))
31, 2bi2bian9 640 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝑥 S 𝐴𝑥 S 𝐵) ↔ (𝑥𝐴𝑥𝐵)))
43albidv 1922 . 2 ((𝐴𝑉𝐵𝑊) → (∀𝑥(𝑥 S 𝐴𝑥 S 𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵)))
5 relssr 35845 . . 3 Rel S
6 releccnveq 35624 . . 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 5334 . 2 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
94, 7, 83bitr4g 317 1 ((𝐴𝑉𝐵𝑊) → ([𝐴] S = [𝐵] S ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wcel 2115  wss 3919   class class class wbr 5052  ccnv 5541  Rel wrel 5547  [cec 8283   S cssr 35561
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ec 8287  df-ssr 35843
This theorem is referenced by: (None)
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