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Theorem preimalegt 39390
Description: The preimage of a left-open, unbounded above interval, is the complement of a right-close, 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 . . 3 𝑥𝜑
2 eldifi 3689 . . . . . . . 8 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) → 𝑥𝐴)
32adantl 480 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝑥𝐴)
42anim1i 589 . . . . . . . . . . 11 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ∧ 𝐵𝐶) → (𝑥𝐴𝐵𝐶))
5 rabid 3090 . . . . . . . . . . 11 (𝑥 ∈ {𝑥𝐴𝐵𝐶} ↔ (𝑥𝐴𝐵𝐶))
64, 5sylibr 222 . . . . . . . . . 10 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ∧ 𝐵𝐶) → 𝑥 ∈ {𝑥𝐴𝐵𝐶})
7 eldifn 3690 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) → ¬ 𝑥 ∈ {𝑥𝐴𝐵𝐶})
87adantr 479 . . . . . . . . . 10 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ∧ 𝐵𝐶) → ¬ 𝑥 ∈ {𝑥𝐴𝐵𝐶})
96, 8pm2.65da 597 . . . . . . . . 9 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) → ¬ 𝐵𝐶)
109adantl 480 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → ¬ 𝐵𝐶)
11 preimalegt.c . . . . . . . . . 10 (𝜑𝐶 ∈ ℝ*)
1211adantr 479 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝐶 ∈ ℝ*)
13 preimalegt.b . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
143, 13syldan 485 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝐵 ∈ ℝ*)
1512, 14xrltnled 38320 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → (𝐶 < 𝐵 ↔ ¬ 𝐵𝐶))
1610, 15mpbird 245 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝐶 < 𝐵)
173, 16jca 552 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → (𝑥𝐴𝐶 < 𝐵))
18 rabid 3090 . . . . . 6 (𝑥 ∈ {𝑥𝐴𝐶 < 𝐵} ↔ (𝑥𝐴𝐶 < 𝐵))
1917, 18sylibr 222 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝑥 ∈ {𝑥𝐴𝐶 < 𝐵})
2019ex 448 . . . 4 (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) → 𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}))
2118simplbi 474 . . . . . . 7 (𝑥 ∈ {𝑥𝐴𝐶 < 𝐵} → 𝑥𝐴)
2221adantl 480 . . . . . 6 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝑥𝐴)
2318simprbi 478 . . . . . . . . . 10 (𝑥 ∈ {𝑥𝐴𝐶 < 𝐵} → 𝐶 < 𝐵)
2423adantl 480 . . . . . . . . 9 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝐶 < 𝐵)
2511adantr 479 . . . . . . . . . 10 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝐶 ∈ ℝ*)
2622, 13syldan 485 . . . . . . . . . 10 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝐵 ∈ ℝ*)
2725, 26xrltnled 38320 . . . . . . . . 9 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → (𝐶 < 𝐵 ↔ ¬ 𝐵𝐶))
2824, 27mpbid 220 . . . . . . . 8 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → ¬ 𝐵𝐶)
2928intnand 952 . . . . . . 7 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → ¬ (𝑥𝐴𝐵𝐶))
3029, 5sylnibr 317 . . . . . 6 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → ¬ 𝑥 ∈ {𝑥𝐴𝐵𝐶})
3122, 30eldifd 3546 . . . . 5 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}))
3231ex 448 . . . 4 (𝜑 → (𝑥 ∈ {𝑥𝐴𝐶 < 𝐵} → 𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})))
3320, 32impbid 200 . . 3 (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ↔ 𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}))
341, 33alrimi 2066 . 2 (𝜑 → ∀𝑥(𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ↔ 𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}))
35 nfcv 2746 . . . 4 𝑥𝐴
36 nfrab1 3094 . . . 4 𝑥{𝑥𝐴𝐵𝐶}
3735, 36nfdif 3688 . . 3 𝑥(𝐴 ∖ {𝑥𝐴𝐵𝐶})
38 nfrab1 3094 . . 3 𝑥{𝑥𝐴𝐶 < 𝐵}
3937, 38dfcleqf 38080 . 2 ((𝐴 ∖ {𝑥𝐴𝐵𝐶}) = {𝑥𝐴𝐶 < 𝐵} ↔ ∀𝑥(𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ↔ 𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}))
4034, 39sylibr 222 1 (𝜑 → (𝐴 ∖ {𝑥𝐴𝐵𝐶}) = {𝑥𝐴𝐶 < 𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wnf 1698  wcel 1975  {crab 2895  cdif 3532   class class class wbr 4573  *cxr 9925   < clt 9926  cle 9927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-br 4574  df-opab 4634  df-xp 5030  df-cnv 5032  df-le 9932
This theorem is referenced by:  salpreimalegt  39397
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