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Theorem cvbr 28987
Description: Binary relation expressing 𝐵 covers 𝐴, which means that 𝐵 is larger than 𝐴 and there is nothing in between. Definition 3.2.18 of [PtakPulmannova] p. 68. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvbr ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem cvbr
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2686 . . . . 5 (𝑦 = 𝐴 → (𝑦C𝐴C ))
21anbi1d 740 . . . 4 (𝑦 = 𝐴 → ((𝑦C𝑧C ) ↔ (𝐴C𝑧C )))
3 psseq1 3672 . . . . 5 (𝑦 = 𝐴 → (𝑦𝑧𝐴𝑧))
4 psseq1 3672 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
54anbi1d 740 . . . . . . 7 (𝑦 = 𝐴 → ((𝑦𝑥𝑥𝑧) ↔ (𝐴𝑥𝑥𝑧)))
65rexbidv 3045 . . . . . 6 (𝑦 = 𝐴 → (∃𝑥C (𝑦𝑥𝑥𝑧) ↔ ∃𝑥C (𝐴𝑥𝑥𝑧)))
76notbid 308 . . . . 5 (𝑦 = 𝐴 → (¬ ∃𝑥C (𝑦𝑥𝑥𝑧) ↔ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧)))
83, 7anbi12d 746 . . . 4 (𝑦 = 𝐴 → ((𝑦𝑧 ∧ ¬ ∃𝑥C (𝑦𝑥𝑥𝑧)) ↔ (𝐴𝑧 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧))))
92, 8anbi12d 746 . . 3 (𝑦 = 𝐴 → (((𝑦C𝑧C ) ∧ (𝑦𝑧 ∧ ¬ ∃𝑥C (𝑦𝑥𝑥𝑧))) ↔ ((𝐴C𝑧C ) ∧ (𝐴𝑧 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧)))))
10 eleq1 2686 . . . . 5 (𝑧 = 𝐵 → (𝑧C𝐵C ))
1110anbi2d 739 . . . 4 (𝑧 = 𝐵 → ((𝐴C𝑧C ) ↔ (𝐴C𝐵C )))
12 psseq2 3673 . . . . 5 (𝑧 = 𝐵 → (𝐴𝑧𝐴𝐵))
13 psseq2 3673 . . . . . . . 8 (𝑧 = 𝐵 → (𝑥𝑧𝑥𝐵))
1413anbi2d 739 . . . . . . 7 (𝑧 = 𝐵 → ((𝐴𝑥𝑥𝑧) ↔ (𝐴𝑥𝑥𝐵)))
1514rexbidv 3045 . . . . . 6 (𝑧 = 𝐵 → (∃𝑥C (𝐴𝑥𝑥𝑧) ↔ ∃𝑥C (𝐴𝑥𝑥𝐵)))
1615notbid 308 . . . . 5 (𝑧 = 𝐵 → (¬ ∃𝑥C (𝐴𝑥𝑥𝑧) ↔ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)))
1712, 16anbi12d 746 . . . 4 (𝑧 = 𝐵 → ((𝐴𝑧 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧)) ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
1811, 17anbi12d 746 . . 3 (𝑧 = 𝐵 → (((𝐴C𝑧C ) ∧ (𝐴𝑧 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧))) ↔ ((𝐴C𝐵C ) ∧ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)))))
19 df-cv 28984 . . 3 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦C𝑧C ) ∧ (𝑦𝑧 ∧ ¬ ∃𝑥C (𝑦𝑥𝑥𝑧)))}
209, 18, 19brabg 4954 . 2 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ ((𝐴C𝐵C ) ∧ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)))))
2120bianabs 923 1 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  wpss 3556   class class class wbr 4613   C cch 27632   ccv 27667
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 4741  ax-nul 4749  ax-pr 4867
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-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-cv 28984
This theorem is referenced by:  cvbr2  28988  cvcon3  28989  cvpss  28990  cvnbtwn  28991
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