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Theorem cvbr 30149
 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 2838 . . . . 5 (𝑦 = 𝐴 → (𝑦C𝐴C ))
21anbi1d 633 . . . 4 (𝑦 = 𝐴 → ((𝑦C𝑧C ) ↔ (𝐴C𝑧C )))
3 psseq1 3989 . . . . 5 (𝑦 = 𝐴 → (𝑦𝑧𝐴𝑧))
4 psseq1 3989 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
54anbi1d 633 . . . . . . 7 (𝑦 = 𝐴 → ((𝑦𝑥𝑥𝑧) ↔ (𝐴𝑥𝑥𝑧)))
65rexbidv 3219 . . . . . 6 (𝑦 = 𝐴 → (∃𝑥C (𝑦𝑥𝑥𝑧) ↔ ∃𝑥C (𝐴𝑥𝑥𝑧)))
76notbid 322 . . . . 5 (𝑦 = 𝐴 → (¬ ∃𝑥C (𝑦𝑥𝑥𝑧) ↔ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧)))
83, 7anbi12d 634 . . . 4 (𝑦 = 𝐴 → ((𝑦𝑧 ∧ ¬ ∃𝑥C (𝑦𝑥𝑥𝑧)) ↔ (𝐴𝑧 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧))))
92, 8anbi12d 634 . . 3 (𝑦 = 𝐴 → (((𝑦C𝑧C ) ∧ (𝑦𝑧 ∧ ¬ ∃𝑥C (𝑦𝑥𝑥𝑧))) ↔ ((𝐴C𝑧C ) ∧ (𝐴𝑧 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧)))))
10 eleq1 2838 . . . . 5 (𝑧 = 𝐵 → (𝑧C𝐵C ))
1110anbi2d 632 . . . 4 (𝑧 = 𝐵 → ((𝐴C𝑧C ) ↔ (𝐴C𝐵C )))
12 psseq2 3990 . . . . 5 (𝑧 = 𝐵 → (𝐴𝑧𝐴𝐵))
13 psseq2 3990 . . . . . . . 8 (𝑧 = 𝐵 → (𝑥𝑧𝑥𝐵))
1413anbi2d 632 . . . . . . 7 (𝑧 = 𝐵 → ((𝐴𝑥𝑥𝑧) ↔ (𝐴𝑥𝑥𝐵)))
1514rexbidv 3219 . . . . . 6 (𝑧 = 𝐵 → (∃𝑥C (𝐴𝑥𝑥𝑧) ↔ ∃𝑥C (𝐴𝑥𝑥𝐵)))
1615notbid 322 . . . . 5 (𝑧 = 𝐵 → (¬ ∃𝑥C (𝐴𝑥𝑥𝑧) ↔ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)))
1712, 16anbi12d 634 . . . 4 (𝑧 = 𝐵 → ((𝐴𝑧 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧)) ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
1811, 17anbi12d 634 . . 3 (𝑧 = 𝐵 → (((𝐴C𝑧C ) ∧ (𝐴𝑧 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧))) ↔ ((𝐴C𝐵C ) ∧ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)))))
19 df-cv 30146 . . 3 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦C𝑧C ) ∧ (𝑦𝑧 ∧ ¬ ∃𝑥C (𝑦𝑥𝑥𝑧)))}
209, 18, 19brabg 5389 . 2 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ ((𝐴C𝐵C ) ∧ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)))))
2120bianabs 546 1 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∃wrex 3069   ⊊ wpss 3855   class class class wbr 5025   Cℋ cch 28796   ⋖ℋ ccv 28831 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pr 5291 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ne 2950  df-rex 3074  df-v 3409  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-pss 3873  df-nul 4222  df-if 4414  df-sn 4516  df-pr 4518  df-op 4522  df-br 5026  df-opab 5088  df-cv 30146 This theorem is referenced by:  cvbr2  30150  cvcon3  30151  cvpss  30152  cvnbtwn  30153
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