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Theorem preimalegt 45901
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 2895 . . 3 𝑥𝐴
3 nfrab1 3443 . . 3 𝑥{𝑥𝐴𝐵𝐶}
42, 3nfdif 4117 . 2 𝑥(𝐴 ∖ {𝑥𝐴𝐵𝐶})
5 nfrab1 3443 . 2 𝑥{𝑥𝐴𝐶 < 𝐵}
6 eldifi 4118 . . . . 5 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) → 𝑥𝐴)
76adantl 481 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝑥𝐴)
8 eldifn 4119 . . . . . . 7 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) → ¬ 𝑥 ∈ {𝑥𝐴𝐵𝐶})
96anim1i 614 . . . . . . . 8 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ∧ 𝐵𝐶) → (𝑥𝐴𝐵𝐶))
10 rabid 3444 . . . . . . . 8 (𝑥 ∈ {𝑥𝐴𝐵𝐶} ↔ (𝑥𝐴𝐵𝐶))
119, 10sylibr 233 . . . . . . 7 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ∧ 𝐵𝐶) → 𝑥 ∈ {𝑥𝐴𝐵𝐶})
128, 11mtand 813 . . . . . 6 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) → ¬ 𝐵𝐶)
1312adantl 481 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → ¬ 𝐵𝐶)
14 preimalegt.c . . . . . . 7 (𝜑𝐶 ∈ ℝ*)
1514adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝐶 ∈ ℝ*)
16 preimalegt.b . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
176, 16sylan2 592 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝐵 ∈ ℝ*)
1815, 17xrltnled 44558 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → (𝐶 < 𝐵 ↔ ¬ 𝐵𝐶))
1913, 18mpbird 257 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝐶 < 𝐵)
20 rabid 3444 . . . 4 (𝑥 ∈ {𝑥𝐴𝐶 < 𝐵} ↔ (𝑥𝐴𝐶 < 𝐵))
217, 19, 20sylanbrc 582 . . 3 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶})) → 𝑥 ∈ {𝑥𝐴𝐶 < 𝐵})
22 rabidim1 3445 . . . . 5 (𝑥 ∈ {𝑥𝐴𝐶 < 𝐵} → 𝑥𝐴)
2322adantl 481 . . . 4 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝑥𝐴)
24 rabidim2 44279 . . . . . . . 8 (𝑥 ∈ {𝑥𝐴𝐶 < 𝐵} → 𝐶 < 𝐵)
2524adantl 481 . . . . . . 7 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝐶 < 𝐵)
2614adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝐶 ∈ ℝ*)
2722, 16sylan2 592 . . . . . . . 8 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝐵 ∈ ℝ*)
2826, 27xrltnled 44558 . . . . . . 7 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → (𝐶 < 𝐵 ↔ ¬ 𝐵𝐶))
2925, 28mpbid 231 . . . . . 6 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → ¬ 𝐵𝐶)
3029intnand 488 . . . . 5 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → ¬ (𝑥𝐴𝐵𝐶))
3130, 10sylnibr 329 . . . 4 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → ¬ 𝑥 ∈ {𝑥𝐴𝐵𝐶})
3223, 31eldifd 3951 . . 3 ((𝜑𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}) → 𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}))
3321, 32impbida 798 . 2 (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐵𝐶}) ↔ 𝑥 ∈ {𝑥𝐴𝐶 < 𝐵}))
341, 4, 5, 33eqrd 3993 1 (𝜑 → (𝐴 ∖ {𝑥𝐴𝐵𝐶}) = {𝑥𝐴𝐶 < 𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1533  wnf 1777  wcel 2098  {crab 3424  cdif 3937   class class class wbr 5138  *cxr 11244   < clt 11245  cle 11246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-cnv 5674  df-le 11251
This theorem is referenced by:  salpreimalegt  45910
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