Theorem preimalegt 46715

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	⊢ Ⅎ𝑥𝜑
preimalegt.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
preimalegt.c	⊢ (𝜑 → 𝐶 ∈ ℝ*)

Assertion

Ref	Expression
preimalegt	⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵})

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem preimalegt

Step	Hyp	Ref	Expression
1		preimalegt.x	. 2 ⊢ Ⅎ𝑥𝜑
2		nfcv 2905	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfrab1 3457	. . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}
4	2, 3	nfdif 4129	. 2 ⊢ Ⅎ𝑥(𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶})
5		nfrab1 3457	. 2 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}
6		eldifi 4131	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) → 𝑥 ∈ 𝐴)
7	6	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶})) → 𝑥 ∈ 𝐴)
8		eldifn 4132	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) → ¬ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶})
9	6	anim1i 615	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) ∧ 𝐵 ≤ 𝐶) → (𝑥 ∈ 𝐴 ∧ 𝐵 ≤ 𝐶))
10		rabid 3458	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ≤ 𝐶))
11	9, 10	sylibr 234	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) ∧ 𝐵 ≤ 𝐶) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶})
12	8, 11	mtand 816	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) → ¬ 𝐵 ≤ 𝐶)
13	12	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶})) → ¬ 𝐵 ≤ 𝐶)
14		preimalegt.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ*)
15	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶})) → 𝐶 ∈ ℝ*)
16		preimalegt.b	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
17	6, 16	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶})) → 𝐵 ∈ ℝ*)
18	15, 17	xrltnled 45374	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶})) → (𝐶 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐶))
19	13, 18	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶})) → 𝐶 < 𝐵)
20		rabid 3458	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 < 𝐵))
21	7, 19, 20	sylanbrc 583	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶})) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵})
22		rabidim1 3459	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} → 𝑥 ∈ 𝐴)
23	22	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}) → 𝑥 ∈ 𝐴)
24		rabidim2 45107	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} → 𝐶 < 𝐵)
25	24	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}) → 𝐶 < 𝐵)
26	14	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}) → 𝐶 ∈ ℝ*)
27	22, 16	sylan2 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}) → 𝐵 ∈ ℝ*)
28	26, 27	xrltnled 45374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}) → (𝐶 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐶))
29	25, 28	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}) → ¬ 𝐵 ≤ 𝐶)
30	29	intnand 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}) → ¬ (𝑥 ∈ 𝐴 ∧ 𝐵 ≤ 𝐶))
31	30, 10	sylnibr 329	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}) → ¬ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶})
32	23, 31	eldifd 3962	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}) → 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}))
33	21, 32	impbida 801	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) ↔ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}))
34	1, 4, 5, 33	eqrd 4003	1 ⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108  {crab 3436   ∖ cdif 3948   class class class wbr 5143  ℝ^*cxr 11294   < clt 11295   ≤ cle 11296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-le 11301

This theorem is referenced by:  salpreimalegt  46724