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Theorem cvbr2 30364
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 30363 . 2 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
2 iman 405 . . . . . 6 (((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵) ↔ ¬ ((𝐴𝑥𝑥𝐵) ∧ ¬ 𝑥 = 𝐵))
3 anass 472 . . . . . . 7 (((𝐴𝑥𝑥𝐵) ∧ ¬ 𝑥 = 𝐵) ↔ (𝐴𝑥 ∧ (𝑥𝐵 ∧ ¬ 𝑥 = 𝐵)))
4 dfpss2 4000 . . . . . . . 8 (𝑥𝐵 ↔ (𝑥𝐵 ∧ ¬ 𝑥 = 𝐵))
54anbi2i 626 . . . . . . 7 ((𝐴𝑥𝑥𝐵) ↔ (𝐴𝑥 ∧ (𝑥𝐵 ∧ ¬ 𝑥 = 𝐵)))
63, 5bitr4i 281 . . . . . 6 (((𝐴𝑥𝑥𝐵) ∧ ¬ 𝑥 = 𝐵) ↔ (𝐴𝑥𝑥𝐵))
72, 6xchbinx 337 . . . . 5 (((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵) ↔ ¬ (𝐴𝑥𝑥𝐵))
87ralbii 3088 . . . 4 (∀𝑥C ((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵) ↔ ∀𝑥C ¬ (𝐴𝑥𝑥𝐵))
9 ralnex 3158 . . . 4 (∀𝑥C ¬ (𝐴𝑥𝑥𝐵) ↔ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))
108, 9bitri 278 . . 3 (∀𝑥C ((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵) ↔ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))
1110anbi2i 626 . 2 ((𝐴𝐵 ∧ ∀𝑥C ((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵)) ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)))
121, 11bitr4di 292 1 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ∀𝑥C ((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wral 3061  wrex 3062  wss 3866  wpss 3867   class class class wbr 5053   C cch 29010   ccv 29045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-cv 30360
This theorem is referenced by:  spansncv2  30374  elat2  30421
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