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Theorem brltc 6114
 Description: Binary relationship form of cardinal less than. (Contributed by SF, 4-Mar-2015.)
Assertion
Ref Expression
brltc (A <c B ↔ (Ac B AB))

Proof of Theorem brltc
StepHypRef Expression
1 brex 4689 . . 3 (A <c B → (A V B V))
21simprd 449 . 2 (A <c BB V)
3 brex 4689 . . . 4 (Ac B → (A V B V))
43simprd 449 . . 3 (Ac BB V)
54adantr 451 . 2 ((Ac B AB) → B V)
6 df-ltc 6100 . . . . 5 <c = ( ≤c I )
76breqi 4645 . . . 4 (A <c BA( ≤c I )B)
8 brdif 4694 . . . 4 (A( ≤c I )B ↔ (Ac B ¬ A I B))
97, 8bitri 240 . . 3 (A <c B ↔ (Ac B ¬ A I B))
10 ideqg 4868 . . . . 5 (B V → (A I BA = B))
1110necon3bbid 2550 . . . 4 (B V → (¬ A I BAB))
1211anbi2d 684 . . 3 (B V → ((Ac B ¬ A I B) ↔ (Ac B AB)))
139, 12syl5bb 248 . 2 (B V → (A <c B ↔ (Ac B AB)))
142, 5, 13pm5.21nii 342 1 (A <c B ↔ (Ac B AB))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   ∈ wcel 1710   ≠ wne 2516  Vcvv 2859   ∖ cdif 3206   class class class wbr 4639   I cid 4763   ≤c clec 6089
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