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Theorem preimalegt 42442
Description: The preimage of a left-open, unbounded above interval, is the complement of a right-closed unbounded below interval. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
preimalegt.x 𝑥𝜑
preimalegt.b ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
preimalegt.c (𝜑𝐶 ∈ ℝ*)
Assertion
Ref Expression
preimalegt (𝜑 → (𝐴 ∖ {𝑥𝐴𝐵𝐶}) = {𝑥𝐴𝐶 < 𝐵})
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem preimalegt
StepHypRef Expression
1 preimalegt.x . 2 𝑥𝜑
2 nfcv 2927 . . 3 𝑥𝐴
3 nfrab1 3319 . . 3 𝑥{𝑥𝐴𝐵𝐶}
42, 3nfdif 3987 . 2 𝑥(𝐴 ∖ {𝑥𝐴𝐵𝐶})
5 nfrab1 3319 . 2 𝑥{𝑥𝐴𝐶 < 𝐵}
6 eldifi 3988 . . . . 5 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) → 𝑥𝐴)
76adantl 474 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝑥𝐴)
8 eldifn 3989 . . . . . . 7 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) → ¬ 𝑥 ∈ {𝑥𝐴𝐵𝐶})
96anim1i 606 . . . . . . . 8 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ∧ 𝐵𝐶) → (𝑥𝐴𝐵𝐶))
10 rabid 3312 . . . . . . . 8 (𝑥 ∈ {𝑥𝐴𝐵𝐶} ↔ (𝑥𝐴𝐵𝐶))
119, 10sylibr 226 . . . . . . 7 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ∧ 𝐵𝐶) → 𝑥 ∈ {𝑥𝐴𝐵𝐶})
128, 11mtand 804 . . . . . 6 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) → ¬ 𝐵𝐶)
1312adantl 474 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → ¬ 𝐵𝐶)
14 preimalegt.c . . . . . . 7 (𝜑𝐶 ∈ ℝ*)
1514adantr 473 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝐶 ∈ ℝ*)
16 preimalegt.b . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
176, 16sylan2 584 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝐵 ∈ ℝ*)
1815, 17xrltnled 41090 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → (𝐶 < 𝐵 ↔ ¬ 𝐵𝐶))
1913, 18mpbird 249 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝐶 < 𝐵)
20 rabid 3312 . . . 4 (𝑥 ∈ {𝑥𝐴𝐶 < 𝐵} ↔ (𝑥𝐴𝐶 < 𝐵))
217, 19, 20sylanbrc 575 . . 3 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝑥 ∈ {𝑥𝐴𝐶 < 𝐵})
22 rabidim1 3314 . . . . 5 (𝑥 ∈ {𝑥𝐴𝐶 < 𝐵} → 𝑥𝐴)
2322adantl 474 . . . 4 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝑥𝐴)
24 rabidim2 40823 . . . . . . . 8 (𝑥 ∈ {𝑥𝐴𝐶 < 𝐵} → 𝐶 < 𝐵)
2524adantl 474 . . . . . . 7 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝐶 < 𝐵)
2614adantr 473 . . . . . . . 8 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝐶 ∈ ℝ*)
2722, 16sylan2 584 . . . . . . . 8 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝐵 ∈ ℝ*)
2826, 27xrltnled 41090 . . . . . . 7 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → (𝐶 < 𝐵 ↔ ¬ 𝐵𝐶))
2925, 28mpbid 224 . . . . . 6 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → ¬ 𝐵𝐶)
3029intnand 481 . . . . 5 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → ¬ (𝑥𝐴𝐵𝐶))
3130, 10sylnibr 321 . . . 4 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → ¬ 𝑥 ∈ {𝑥𝐴𝐵𝐶})
3223, 31eldifd 3835 . . 3 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}))
3321, 32impbida 789 . 2 (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ↔ 𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}))
341, 4, 5, 33eqrd 3872 1 (𝜑 → (𝐴 ∖ {𝑥𝐴𝐵𝐶}) = {𝑥𝐴𝐶 < 𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387   = wceq 1508  wnf 1747  wcel 2051  {crab 3087  cdif 3821   class class class wbr 4926  *cxr 10472   < clt 10473  cle 10474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pr 5183
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-op 4443  df-br 4927  df-opab 4989  df-xp 5410  df-cnv 5412  df-le 10479
This theorem is referenced by:  salpreimalegt  42449
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