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Theorem cvbr2 29012
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 29011 . 2 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
2 iman 440 . . . . . 6 (((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵) ↔ ¬ ((𝐴𝑥𝑥𝐵) ∧ ¬ 𝑥 = 𝐵))
3 anass 680 . . . . . . 7 (((𝐴𝑥𝑥𝐵) ∧ ¬ 𝑥 = 𝐵) ↔ (𝐴𝑥 ∧ (𝑥𝐵 ∧ ¬ 𝑥 = 𝐵)))
4 dfpss2 3675 . . . . . . . 8 (𝑥𝐵 ↔ (𝑥𝐵 ∧ ¬ 𝑥 = 𝐵))
54anbi2i 729 . . . . . . 7 ((𝐴𝑥𝑥𝐵) ↔ (𝐴𝑥 ∧ (𝑥𝐵 ∧ ¬ 𝑥 = 𝐵)))
63, 5bitr4i 267 . . . . . 6 (((𝐴𝑥𝑥𝐵) ∧ ¬ 𝑥 = 𝐵) ↔ (𝐴𝑥𝑥𝐵))
72, 6xchbinx 324 . . . . 5 (((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵) ↔ ¬ (𝐴𝑥𝑥𝐵))
87ralbii 2975 . . . 4 (∀𝑥C ((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵) ↔ ∀𝑥C ¬ (𝐴𝑥𝑥𝐵))
9 ralnex 2987 . . . 4 (∀𝑥C ¬ (𝐴𝑥𝑥𝐵) ↔ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))
108, 9bitri 264 . . 3 (∀𝑥C ((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵) ↔ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))
1110anbi2i 729 . 2 ((𝐴𝐵 ∧ ∀𝑥C ((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵)) ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)))
121, 11syl6bbr 278 1 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ∀𝑥C ((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  wss 3559  wpss 3560   class class class wbr 4618   C cch 27656   ccv 27691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-cv 29008
This theorem is referenced by:  spansncv2  29022  elat2  29069
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