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Theorem preimalegt 40676
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 3724 . . . . . . . 8 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) → 𝑥𝐴)
32adantl 482 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝑥𝐴)
42anim1i 591 . . . . . . . . . . 11 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ∧ 𝐵𝐶) → (𝑥𝐴𝐵𝐶))
5 rabid 3111 . . . . . . . . . . 11 (𝑥 ∈ {𝑥𝐴𝐵𝐶} ↔ (𝑥𝐴𝐵𝐶))
64, 5sylibr 224 . . . . . . . . . 10 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ∧ 𝐵𝐶) → 𝑥 ∈ {𝑥𝐴𝐵𝐶})
7 eldifn 3725 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) → ¬ 𝑥 ∈ {𝑥𝐴𝐵𝐶})
87adantr 481 . . . . . . . . . 10 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ∧ 𝐵𝐶) → ¬ 𝑥 ∈ {𝑥𝐴𝐵𝐶})
96, 8pm2.65da 599 . . . . . . . . 9 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) → ¬ 𝐵𝐶)
109adantl 482 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → ¬ 𝐵𝐶)
11 preimalegt.c . . . . . . . . . 10 (𝜑𝐶 ∈ ℝ*)
1211adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝐶 ∈ ℝ*)
13 preimalegt.b . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
143, 13syldan 487 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝐵 ∈ ℝ*)
1512, 14xrltnled 39392 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → (𝐶 < 𝐵 ↔ ¬ 𝐵𝐶))
1610, 15mpbird 247 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝐶 < 𝐵)
173, 16jca 554 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → (𝑥𝐴𝐶 < 𝐵))
18 rabid 3111 . . . . . 6 (𝑥 ∈ {𝑥𝐴𝐶 < 𝐵} ↔ (𝑥𝐴𝐶 < 𝐵))
1917, 18sylibr 224 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝑥 ∈ {𝑥𝐴𝐶 < 𝐵})
2019ex 450 . . . 4 (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) → 𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}))
2118simplbi 476 . . . . . . 7 (𝑥 ∈ {𝑥𝐴𝐶 < 𝐵} → 𝑥𝐴)
2221adantl 482 . . . . . 6 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝑥𝐴)
2318simprbi 480 . . . . . . . . . 10 (𝑥 ∈ {𝑥𝐴𝐶 < 𝐵} → 𝐶 < 𝐵)
2423adantl 482 . . . . . . . . 9 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝐶 < 𝐵)
2511adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝐶 ∈ ℝ*)
2622, 13syldan 487 . . . . . . . . . 10 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝐵 ∈ ℝ*)
2725, 26xrltnled 39392 . . . . . . . . 9 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → (𝐶 < 𝐵 ↔ ¬ 𝐵𝐶))
2824, 27mpbid 222 . . . . . . . 8 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → ¬ 𝐵𝐶)
2928intnand 961 . . . . . . 7 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → ¬ (𝑥𝐴𝐵𝐶))
3029, 5sylnibr 319 . . . . . 6 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → ¬ 𝑥 ∈ {𝑥𝐴𝐵𝐶})
3122, 30eldifd 3578 . . . . 5 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}))
3231ex 450 . . . 4 (𝜑 → (𝑥 ∈ {𝑥𝐴𝐶 < 𝐵} → 𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})))
3320, 32impbid 202 . . 3 (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ↔ 𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}))
341, 33alrimi 2080 . 2 (𝜑 → ∀𝑥(𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ↔ 𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}))
35 nfcv 2762 . . . 4 𝑥𝐴
36 nfrab1 3117 . . . 4 𝑥{𝑥𝐴𝐵𝐶}
3735, 36nfdif 3723 . . 3 𝑥(𝐴 ∖ {𝑥𝐴𝐵𝐶})
38 nfrab1 3117 . . 3 𝑥{𝑥𝐴𝐶 < 𝐵}
3937, 38dfcleqf 39075 . 2 ((𝐴 ∖ {𝑥𝐴𝐵𝐶}) = {𝑥𝐴𝐶 < 𝐵} ↔ ∀𝑥(𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ↔ 𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}))
4034, 39sylibr 224 1 (𝜑 → (𝐴 ∖ {𝑥𝐴𝐵𝐶}) = {𝑥𝐴𝐶 < 𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1479   = wceq 1481  wnf 1706  wcel 1988  {crab 2913  cdif 3564   class class class wbr 4644  *cxr 10058   < clt 10059  cle 10060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-cnv 5112  df-le 10065
This theorem is referenced by:  salpreimalegt  40683
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