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Theorem cossss 35151
 Description: Subclass theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 11-Nov-2019.)
Assertion
Ref Expression
cossss (𝐴𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)

Proof of Theorem cossss
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssbr 5000 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑦𝑥𝐵𝑦))
2 ssbr 5000 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑧𝑥𝐵𝑧))
31, 2anim12d 608 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝑦𝑥𝐴𝑧) → (𝑥𝐵𝑦𝑥𝐵𝑧)))
43eximdv 1893 . . 3 (𝐴𝐵 → (∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧) → ∃𝑥(𝑥𝐵𝑦𝑥𝐵𝑧)))
54ssopab2dv 5318 . 2 (𝐴𝐵 → {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐵𝑦𝑥𝐵𝑧)})
6 df-coss 35140 . 2 𝐴 = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧)}
7 df-coss 35140 . 2 𝐵 = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐵𝑦𝑥𝐵𝑧)}
85, 6, 73sstr4g 3928 1 (𝐴𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396  ∃wex 1759   ⊆ wss 3854   class class class wbr 4956  {copab 5018   ≀ ccoss 34933 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-in 3861  df-ss 3869  df-br 4957  df-opab 5019  df-coss 35140 This theorem is referenced by:  funALTVss  35413
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