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Theorem cvbr2 31267
Description: Binary relation expressing 𝐵 covers 𝐴. Definition of covers in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvbr2 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ∀𝑥C ((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem cvbr2
StepHypRef Expression
1 cvbr 31266 . 2 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
2 iman 403 . . . . . 6 (((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵) ↔ ¬ ((𝐴𝑥𝑥𝐵) ∧ ¬ 𝑥 = 𝐵))
3 anass 470 . . . . . . 7 (((𝐴𝑥𝑥𝐵) ∧ ¬ 𝑥 = 𝐵) ↔ (𝐴𝑥 ∧ (𝑥𝐵 ∧ ¬ 𝑥 = 𝐵)))
4 dfpss2 4050 . . . . . . . 8 (𝑥𝐵 ↔ (𝑥𝐵 ∧ ¬ 𝑥 = 𝐵))
54anbi2i 624 . . . . . . 7 ((𝐴𝑥𝑥𝐵) ↔ (𝐴𝑥 ∧ (𝑥𝐵 ∧ ¬ 𝑥 = 𝐵)))
63, 5bitr4i 278 . . . . . 6 (((𝐴𝑥𝑥𝐵) ∧ ¬ 𝑥 = 𝐵) ↔ (𝐴𝑥𝑥𝐵))
72, 6xchbinx 334 . . . . 5 (((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵) ↔ ¬ (𝐴𝑥𝑥𝐵))
87ralbii 3097 . . . 4 (∀𝑥C ((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵) ↔ ∀𝑥C ¬ (𝐴𝑥𝑥𝐵))
9 ralnex 3076 . . . 4 (∀𝑥C ¬ (𝐴𝑥𝑥𝐵) ↔ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))
108, 9bitri 275 . . 3 (∀𝑥C ((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵) ↔ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))
1110anbi2i 624 . 2 ((𝐴𝐵 ∧ ∀𝑥C ((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵)) ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)))
121, 11bitr4di 289 1 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ∀𝑥C ((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3065  wrex 3074  wss 3915  wpss 3916   class class class wbr 5110   C cch 29913   ccv 29948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-cv 31263
This theorem is referenced by:  spansncv2  31277  elat2  31324
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