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Theorem cvbr 29992
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 2905 . . . . 5 (𝑦 = 𝐴 → (𝑦C𝐴C ))
21anbi1d 629 . . . 4 (𝑦 = 𝐴 → ((𝑦C𝑧C ) ↔ (𝐴C𝑧C )))
3 psseq1 4068 . . . . 5 (𝑦 = 𝐴 → (𝑦𝑧𝐴𝑧))
4 psseq1 4068 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
54anbi1d 629 . . . . . . 7 (𝑦 = 𝐴 → ((𝑦𝑥𝑥𝑧) ↔ (𝐴𝑥𝑥𝑧)))
65rexbidv 3302 . . . . . 6 (𝑦 = 𝐴 → (∃𝑥C (𝑦𝑥𝑥𝑧) ↔ ∃𝑥C (𝐴𝑥𝑥𝑧)))
76notbid 319 . . . . 5 (𝑦 = 𝐴 → (¬ ∃𝑥C (𝑦𝑥𝑥𝑧) ↔ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧)))
83, 7anbi12d 630 . . . 4 (𝑦 = 𝐴 → ((𝑦𝑧 ∧ ¬ ∃𝑥C (𝑦𝑥𝑥𝑧)) ↔ (𝐴𝑧 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧))))
92, 8anbi12d 630 . . 3 (𝑦 = 𝐴 → (((𝑦C𝑧C ) ∧ (𝑦𝑧 ∧ ¬ ∃𝑥C (𝑦𝑥𝑥𝑧))) ↔ ((𝐴C𝑧C ) ∧ (𝐴𝑧 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧)))))
10 eleq1 2905 . . . . 5 (𝑧 = 𝐵 → (𝑧C𝐵C ))
1110anbi2d 628 . . . 4 (𝑧 = 𝐵 → ((𝐴C𝑧C ) ↔ (𝐴C𝐵C )))
12 psseq2 4069 . . . . 5 (𝑧 = 𝐵 → (𝐴𝑧𝐴𝐵))
13 psseq2 4069 . . . . . . . 8 (𝑧 = 𝐵 → (𝑥𝑧𝑥𝐵))
1413anbi2d 628 . . . . . . 7 (𝑧 = 𝐵 → ((𝐴𝑥𝑥𝑧) ↔ (𝐴𝑥𝑥𝐵)))
1514rexbidv 3302 . . . . . 6 (𝑧 = 𝐵 → (∃𝑥C (𝐴𝑥𝑥𝑧) ↔ ∃𝑥C (𝐴𝑥𝑥𝐵)))
1615notbid 319 . . . . 5 (𝑧 = 𝐵 → (¬ ∃𝑥C (𝐴𝑥𝑥𝑧) ↔ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)))
1712, 16anbi12d 630 . . . 4 (𝑧 = 𝐵 → ((𝐴𝑧 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧)) ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
1811, 17anbi12d 630 . . 3 (𝑧 = 𝐵 → (((𝐴C𝑧C ) ∧ (𝐴𝑧 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝑧))) ↔ ((𝐴C𝐵C ) ∧ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)))))
19 df-cv 29989 . . 3 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦C𝑧C ) ∧ (𝑦𝑧 ∧ ¬ ∃𝑥C (𝑦𝑥𝑥𝑧)))}
209, 18, 19brabg 5423 . 2 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ ((𝐴C𝐵C ) ∧ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)))))
2120bianabs 542 1 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wrex 3144  wpss 3941   class class class wbr 5063   C cch 28639   ccv 28674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-cv 29989
This theorem is referenced by:  cvbr2  29993  cvcon3  29994  cvpss  29995  cvnbtwn  29996
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