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Theorem cvbr2 29476
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 29475 . 2 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
2 iman 390 . . . . . 6 (((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵) ↔ ¬ ((𝐴𝑥𝑥𝐵) ∧ ¬ 𝑥 = 𝐵))
3 anass 456 . . . . . . 7 (((𝐴𝑥𝑥𝐵) ∧ ¬ 𝑥 = 𝐵) ↔ (𝐴𝑥 ∧ (𝑥𝐵 ∧ ¬ 𝑥 = 𝐵)))
4 dfpss2 3897 . . . . . . . 8 (𝑥𝐵 ↔ (𝑥𝐵 ∧ ¬ 𝑥 = 𝐵))
54anbi2i 611 . . . . . . 7 ((𝐴𝑥𝑥𝐵) ↔ (𝐴𝑥 ∧ (𝑥𝐵 ∧ ¬ 𝑥 = 𝐵)))
63, 5bitr4i 269 . . . . . 6 (((𝐴𝑥𝑥𝐵) ∧ ¬ 𝑥 = 𝐵) ↔ (𝐴𝑥𝑥𝐵))
72, 6xchbinx 325 . . . . 5 (((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵) ↔ ¬ (𝐴𝑥𝑥𝐵))
87ralbii 3175 . . . 4 (∀𝑥C ((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵) ↔ ∀𝑥C ¬ (𝐴𝑥𝑥𝐵))
9 ralnex 3187 . . . 4 (∀𝑥C ¬ (𝐴𝑥𝑥𝐵) ↔ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))
108, 9bitri 266 . . 3 (∀𝑥C ((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵) ↔ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))
1110anbi2i 611 . 2 ((𝐴𝐵 ∧ ∀𝑥C ((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵)) ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)))
121, 11syl6bbr 280 1 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ∀𝑥C ((𝐴𝑥𝑥𝐵) → 𝑥 = 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  wral 3103  wrex 3104  wss 3776  wpss 3777   class class class wbr 4851   C cch 28120   ccv 28155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pr 5103
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-br 4852  df-opab 4914  df-cv 29472
This theorem is referenced by:  spansncv2  29486  elat2  29533
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