Theorem ssfz12 41324

Description: Subset relationship for finite sets of sequential integers. (Contributed by Alexander van der Vekens, 16-Mar-2018.)

Assertion

Ref	Expression
ssfz12	⊢ ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → ((𝐾...𝐿) ⊆ (𝑀...𝑁) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁)))

Proof of Theorem ssfz12

Step	Hyp	Ref	Expression
1		eluz 11701	. . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐿 ∈ (ℤ≥‘𝐾) ↔ 𝐾 ≤ 𝐿))
2	1	biimp3ar 1433	. . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → 𝐿 ∈ (ℤ≥‘𝐾))
3		eluzfz1 12348	. . 3 ⊢ (𝐿 ∈ (ℤ≥‘𝐾) → 𝐾 ∈ (𝐾...𝐿))
4	2, 3	syl 17	. 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → 𝐾 ∈ (𝐾...𝐿))
5		eluzfz2 12349	. . . 4 ⊢ (𝐿 ∈ (ℤ≥‘𝐾) → 𝐿 ∈ (𝐾...𝐿))
6	2, 5	syl 17	. . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → 𝐿 ∈ (𝐾...𝐿))
7		ssel2 3598	. . . . . . . 8 ⊢ (((𝐾...𝐿) ⊆ (𝑀...𝑁) ∧ 𝐾 ∈ (𝐾...𝐿)) → 𝐾 ∈ (𝑀...𝑁))
8		ssel2 3598	. . . . . . . . . . 11 ⊢ (((𝐾...𝐿) ⊆ (𝑀...𝑁) ∧ 𝐿 ∈ (𝐾...𝐿)) → 𝐿 ∈ (𝑀...𝑁))
9		elfzuz3 12339	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐿))
10		eluz2 11693	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾))
11		eluz2 11693	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ≥‘𝐿) ↔ (𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐿 ≤ 𝑁))
12		pm3.21 464	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 ≤ 𝑁 → (𝑀 ≤ 𝐾 → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁)))
13	12	3ad2ant3 1084	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐿 ≤ 𝑁) → (𝑀 ≤ 𝐾 → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁)))
14	11, 13	sylbi 207	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ≥‘𝐿) → (𝑀 ≤ 𝐾 → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁)))
15	14	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → (𝑁 ∈ (ℤ≥‘𝐿) → (𝑀 ≤ 𝐾 → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁))))
16	15	com13 88	. . . . . . . . . . . . . 14 ⊢ (𝑀 ≤ 𝐾 → (𝑁 ∈ (ℤ≥‘𝐿) → ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁))))
17	16	3ad2ant3 1084	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾) → (𝑁 ∈ (ℤ≥‘𝐿) → ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁))))
18	10, 17	sylbi 207	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝑁 ∈ (ℤ≥‘𝐿) → ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁))))
19		elfzuz 12338	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀))
20	18, 19	syl11 33	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ≥‘𝐿) → (𝐾 ∈ (𝑀...𝑁) → ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁))))
21	8, 9, 20	3syl 18	. . . . . . . . . 10 ⊢ (((𝐾...𝐿) ⊆ (𝑀...𝑁) ∧ 𝐿 ∈ (𝐾...𝐿)) → (𝐾 ∈ (𝑀...𝑁) → ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁))))
22	21	ex 450	. . . . . . . . 9 ⊢ ((𝐾...𝐿) ⊆ (𝑀...𝑁) → (𝐿 ∈ (𝐾...𝐿) → (𝐾 ∈ (𝑀...𝑁) → ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁)))))
23	22	com4t 93	. . . . . . . 8 ⊢ (𝐾 ∈ (𝑀...𝑁) → ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → ((𝐾...𝐿) ⊆ (𝑀...𝑁) → (𝐿 ∈ (𝐾...𝐿) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁)))))
24	7, 23	syl 17	. . . . . . 7 ⊢ (((𝐾...𝐿) ⊆ (𝑀...𝑁) ∧ 𝐾 ∈ (𝐾...𝐿)) → ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → ((𝐾...𝐿) ⊆ (𝑀...𝑁) → (𝐿 ∈ (𝐾...𝐿) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁)))))
25	24	ex 450	. . . . . 6 ⊢ ((𝐾...𝐿) ⊆ (𝑀...𝑁) → (𝐾 ∈ (𝐾...𝐿) → ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → ((𝐾...𝐿) ⊆ (𝑀...𝑁) → (𝐿 ∈ (𝐾...𝐿) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁))))))
26	25	com24 95	. . . . 5 ⊢ ((𝐾...𝐿) ⊆ (𝑀...𝑁) → ((𝐾...𝐿) ⊆ (𝑀...𝑁) → ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → (𝐾 ∈ (𝐾...𝐿) → (𝐿 ∈ (𝐾...𝐿) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁))))))
27	26	pm2.43i 52	. . . 4 ⊢ ((𝐾...𝐿) ⊆ (𝑀...𝑁) → ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → (𝐾 ∈ (𝐾...𝐿) → (𝐿 ∈ (𝐾...𝐿) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁)))))
28	27	com14 96	. . 3 ⊢ (𝐿 ∈ (𝐾...𝐿) → ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → (𝐾 ∈ (𝐾...𝐿) → ((𝐾...𝐿) ⊆ (𝑀...𝑁) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁)))))
29	6, 28	mpcom 38	. 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → (𝐾 ∈ (𝐾...𝐿) → ((𝐾...𝐿) ⊆ (𝑀...𝑁) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁))))
30	4, 29	mpd 15	1 ⊢ ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → ((𝐾...𝐿) ⊆ (𝑀...𝑁) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁)))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   ∈ wcel 1990   ⊆ wss 3574   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650   ≤ cle 10075  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-neg 10269  df-z 11378  df-uz 11688  df-fz 12327

This theorem is referenced by: (None)