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Theorem cvbr 29481
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 615 . . . 4 (𝑦 = 𝐴 → ((𝑦C𝑧C ) ↔ (𝐴C𝑧C )))
3 psseq1 3844 . . . . 5 (𝑦 = 𝐴 → (𝑦𝑧𝐴𝑧))
4 psseq1 3844 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
54anbi1d 615 . . . . . . 7 (𝑦 = 𝐴 → ((𝑦𝑥𝑥𝑧) ↔ (𝐴𝑥𝑥𝑧)))
65rexbidv 3200 . . . . . 6 (𝑦 = 𝐴 → (∃𝑥C (𝑦𝑥𝑥𝑧) ↔ ∃𝑥C (𝐴𝑥𝑥𝑧)))
76notbid 307 . . . . 5 (𝑦 = 𝐴 → (¬ ∃𝑥C (𝑦𝑥𝑥𝑧) ↔ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧)))
83, 7anbi12d 616 . . . 4 (𝑦 = 𝐴 → ((𝑦𝑧 ∧ ¬ ∃𝑥C (𝑦𝑥𝑥𝑧)) ↔ (𝐴𝑧 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧))))
92, 8anbi12d 616 . . 3 (𝑦 = 𝐴 → (((𝑦C𝑧C ) ∧ (𝑦𝑧 ∧ ¬ ∃𝑥C (𝑦𝑥𝑥𝑧))) ↔ ((𝐴C𝑧C ) ∧ (𝐴𝑧 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧)))))
10 eleq1 2838 . . . . 5 (𝑧 = 𝐵 → (𝑧C𝐵C ))
1110anbi2d 614 . . . 4 (𝑧 = 𝐵 → ((𝐴C𝑧C ) ↔ (𝐴C𝐵C )))
12 psseq2 3845 . . . . 5 (𝑧 = 𝐵 → (𝐴𝑧𝐴𝐵))
13 psseq2 3845 . . . . . . . 8 (𝑧 = 𝐵 → (𝑥𝑧𝑥𝐵))
1413anbi2d 614 . . . . . . 7 (𝑧 = 𝐵 → ((𝐴𝑥𝑥𝑧) ↔ (𝐴𝑥𝑥𝐵)))
1514rexbidv 3200 . . . . . 6 (𝑧 = 𝐵 → (∃𝑥C (𝐴𝑥𝑥𝑧) ↔ ∃𝑥C (𝐴𝑥𝑥𝐵)))
1615notbid 307 . . . . 5 (𝑧 = 𝐵 → (¬ ∃𝑥C (𝐴𝑥𝑥𝑧) ↔ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)))
1712, 16anbi12d 616 . . . 4 (𝑧 = 𝐵 → ((𝐴𝑧 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧)) ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
1811, 17anbi12d 616 . . 3 (𝑧 = 𝐵 → (((𝐴C𝑧C ) ∧ (𝐴𝑧 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧))) ↔ ((𝐴C𝐵C ) ∧ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)))))
19 df-cv 29478 . . 3 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦C𝑧C ) ∧ (𝑦𝑧 ∧ ¬ ∃𝑥C (𝑦𝑥𝑥𝑧)))}
209, 18, 19brabg 5127 . 2 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ ((𝐴C𝐵C ) ∧ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)))))
2120bianabs 531 1 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wrex 3062  wpss 3724   class class class wbr 4786   C cch 28126   ccv 28161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-cv 29478
This theorem is referenced by:  cvbr2  29482  cvcon3  29483  cvpss  29484  cvnbtwn  29485
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