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Theorem preimalegt 41234
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 3765 . . . . . . . 8 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) → 𝑥𝐴)
32adantl 481 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝑥𝐴)
42anim1i 591 . . . . . . . . . . 11 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ∧ 𝐵𝐶) → (𝑥𝐴𝐵𝐶))
5 rabid 3145 . . . . . . . . . . 11 (𝑥 ∈ {𝑥𝐴𝐵𝐶} ↔ (𝑥𝐴𝐵𝐶))
64, 5sylibr 224 . . . . . . . . . 10 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ∧ 𝐵𝐶) → 𝑥 ∈ {𝑥𝐴𝐵𝐶})
7 eldifn 3766 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) → ¬ 𝑥 ∈ {𝑥𝐴𝐵𝐶})
87adantr 480 . . . . . . . . . 10 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ∧ 𝐵𝐶) → ¬ 𝑥 ∈ {𝑥𝐴𝐵𝐶})
96, 8pm2.65da 599 . . . . . . . . 9 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) → ¬ 𝐵𝐶)
109adantl 481 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → ¬ 𝐵𝐶)
11 preimalegt.c . . . . . . . . . 10 (𝜑𝐶 ∈ ℝ*)
1211adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝐶 ∈ ℝ*)
13 preimalegt.b . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
143, 13syldan 486 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝐵 ∈ ℝ*)
1512, 14xrltnled 39892 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → (𝐶 < 𝐵 ↔ ¬ 𝐵𝐶))
1610, 15mpbird 247 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝐶 < 𝐵)
173, 16jca 553 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → (𝑥𝐴𝐶 < 𝐵))
18 rabid 3145 . . . . . 6 (𝑥 ∈ {𝑥𝐴𝐶 < 𝐵} ↔ (𝑥𝐴𝐶 < 𝐵))
1917, 18sylibr 224 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝑥 ∈ {𝑥𝐴𝐶 < 𝐵})
2019ex 449 . . . 4 (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) → 𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}))
2118simplbi 475 . . . . . . 7 (𝑥 ∈ {𝑥𝐴𝐶 < 𝐵} → 𝑥𝐴)
2221adantl 481 . . . . . 6 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝑥𝐴)
2318simprbi 479 . . . . . . . . . 10 (𝑥 ∈ {𝑥𝐴𝐶 < 𝐵} → 𝐶 < 𝐵)
2423adantl 481 . . . . . . . . 9 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝐶 < 𝐵)
2511adantr 480 . . . . . . . . . 10 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝐶 ∈ ℝ*)
2622, 13syldan 486 . . . . . . . . . 10 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝐵 ∈ ℝ*)
2725, 26xrltnled 39892 . . . . . . . . 9 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → (𝐶 < 𝐵 ↔ ¬ 𝐵𝐶))
2824, 27mpbid 222 . . . . . . . 8 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → ¬ 𝐵𝐶)
2928intnand 982 . . . . . . 7 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → ¬ (𝑥𝐴𝐵𝐶))
3029, 5sylnibr 318 . . . . . 6 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → ¬ 𝑥 ∈ {𝑥𝐴𝐵𝐶})
3122, 30eldifd 3618 . . . . 5 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}))
3231ex 449 . . . 4 (𝜑 → (𝑥 ∈ {𝑥𝐴𝐶 < 𝐵} → 𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})))
3320, 32impbid 202 . . 3 (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ↔ 𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}))
341, 33alrimi 2120 . 2 (𝜑 → ∀𝑥(𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ↔ 𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}))
35 nfcv 2793 . . . 4 𝑥𝐴
36 nfrab1 3152 . . . 4 𝑥{𝑥𝐴𝐵𝐶}
3735, 36nfdif 3764 . . 3 𝑥(𝐴 ∖ {𝑥𝐴𝐵𝐶})
38 nfrab1 3152 . . 3 𝑥{𝑥𝐴𝐶 < 𝐵}
3937, 38dfcleqf 39569 . 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 383  wal 1521   = wceq 1523  wnf 1748  wcel 2030  {crab 2945  cdif 3604   class class class wbr 4685  *cxr 10111   < clt 10112  cle 10113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-le 10118
This theorem is referenced by:  salpreimalegt  41241
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