Theorem nmembers1 6272

Description: Count the number of elements in a natural interval. From nmembers1lem2 6270 and nmembers1lem3 6271, we would expect to arrive at {m ∈ Nn ∣ (1_c ≤_c m ∧ m ≤_c N)} ∈ N, but this proposition is not stratifiable. Instead, we arrive at the weaker conclusion below. We can arrive at the earlier proposition once we add the Axiom of Counting, which we will do later. (Contributed by Scott Fenton, 3-Aug-2019.)

Assertion

Ref	Expression
nmembers1	⊢ (N ∈ Nn → {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c N)} ∈ Tc Tc N)

Distinct variable group:   m,N

Proof of Theorem nmembers1

Dummy variables n a are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmembers1lem1 6269	. 2 ⊢ {n ∣ {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c n)} ∈ Tc Tc n} ∈ V
2		breq2 4644	. . . . 5 ⊢ (n = 0c → (m ≤c n ↔ m ≤c 0c))
3	2	anbi2d 684	. . . 4 ⊢ (n = 0c → ((1c ≤c m ∧ m ≤c n) ↔ (1c ≤c m ∧ m ≤c 0c)))
4	3	rabbidv 2852	. . 3 ⊢ (n = 0c → {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c n)} = {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c 0c)})
5		tceq 6159	. . . 4 ⊢ (n = 0c → Tc n = Tc 0c)
6		tceq 6159	. . . 4 ⊢ ( Tc n = Tc 0c → Tc Tc n = Tc Tc 0c)
7	5, 6	syl 15	. . 3 ⊢ (n = 0c → Tc Tc n = Tc Tc 0c)
8	4, 7	eleq12d 2421	. 2 ⊢ (n = 0c → ({m ∈ Nn ∣ (1c ≤c m ∧ m ≤c n)} ∈ Tc Tc n ↔ {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c 0c)} ∈ Tc Tc 0c))
9		breq2 4644	. . . . 5 ⊢ (n = a → (m ≤c n ↔ m ≤c a))
10	9	anbi2d 684	. . . 4 ⊢ (n = a → ((1c ≤c m ∧ m ≤c n) ↔ (1c ≤c m ∧ m ≤c a)))
11	10	rabbidv 2852	. . 3 ⊢ (n = a → {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c n)} = {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c a)})
12		tceq 6159	. . . 4 ⊢ (n = a → Tc n = Tc a)
13		tceq 6159	. . . 4 ⊢ ( Tc n = Tc a → Tc Tc n = Tc Tc a)
14	12, 13	syl 15	. . 3 ⊢ (n = a → Tc Tc n = Tc Tc a)
15	11, 14	eleq12d 2421	. 2 ⊢ (n = a → ({m ∈ Nn ∣ (1c ≤c m ∧ m ≤c n)} ∈ Tc Tc n ↔ {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c a)} ∈ Tc Tc a))
16		breq2 4644	. . . . 5 ⊢ (n = (a +c 1c) → (m ≤c n ↔ m ≤c (a +c 1c)))
17	16	anbi2d 684	. . . 4 ⊢ (n = (a +c 1c) → ((1c ≤c m ∧ m ≤c n) ↔ (1c ≤c m ∧ m ≤c (a +c 1c))))
18	17	rabbidv 2852	. . 3 ⊢ (n = (a +c 1c) → {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c n)} = {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c (a +c 1c))})
19		tceq 6159	. . . 4 ⊢ (n = (a +c 1c) → Tc n = Tc (a +c 1c))
20		tceq 6159	. . . 4 ⊢ ( Tc n = Tc (a +c 1c) → Tc Tc n = Tc Tc (a +c 1c))
21	19, 20	syl 15	. . 3 ⊢ (n = (a +c 1c) → Tc Tc n = Tc Tc (a +c 1c))
22	18, 21	eleq12d 2421	. 2 ⊢ (n = (a +c 1c) → ({m ∈ Nn ∣ (1c ≤c m ∧ m ≤c n)} ∈ Tc Tc n ↔ {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c (a +c 1c))} ∈ Tc Tc (a +c 1c)))
23		breq2 4644	. . . . 5 ⊢ (n = N → (m ≤c n ↔ m ≤c N))
24	23	anbi2d 684	. . . 4 ⊢ (n = N → ((1c ≤c m ∧ m ≤c n) ↔ (1c ≤c m ∧ m ≤c N)))
25	24	rabbidv 2852	. . 3 ⊢ (n = N → {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c n)} = {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c N)})
26		tceq 6159	. . . 4 ⊢ (n = N → Tc n = Tc N)
27		tceq 6159	. . . 4 ⊢ ( Tc n = Tc N → Tc Tc n = Tc Tc N)
28	26, 27	syl 15	. . 3 ⊢ (n = N → Tc Tc n = Tc Tc N)
29	25, 28	eleq12d 2421	. 2 ⊢ (n = N → ({m ∈ Nn ∣ (1c ≤c m ∧ m ≤c n)} ∈ Tc Tc n ↔ {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c N)} ∈ Tc Tc N))
30		nmembers1lem2 6270	. . 3 ⊢ {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c 0c)} ∈ 0c
31		tc0c 6164	. . . . 5 ⊢ Tc 0c = 0c
32		tceq 6159	. . . . 5 ⊢ ( Tc 0c = 0c → Tc Tc 0c = Tc 0c)
33	31, 32	ax-mp 5	. . . 4 ⊢ Tc Tc 0c = Tc 0c
34	33, 31	eqtri 2373	. . 3 ⊢ Tc Tc 0c = 0c
35	30, 34	eleqtrri 2426	. 2 ⊢ {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c 0c)} ∈ Tc Tc 0c
36		nntccl 6171	. . . 4 ⊢ (a ∈ Nn → Tc a ∈ Nn )
37		nntccl 6171	. . . 4 ⊢ ( Tc a ∈ Nn → Tc Tc a ∈ Nn )
38	36, 37	syl 15	. . 3 ⊢ (a ∈ Nn → Tc Tc a ∈ Nn )
39		nmembers1lem3 6271	. . . 4 ⊢ ((a ∈ Nn ∧ Tc Tc a ∈ Nn ) → ({m ∈ Nn ∣ (1c ≤c m ∧ m ≤c a)} ∈ Tc Tc a → {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c (a +c 1c))} ∈ ( Tc Tc a +c 1c)))
40		nnnc 6147	. . . . . . 7 ⊢ (a ∈ Nn → a ∈ NC )
41		1cnc 6140	. . . . . . . . . . 11 ⊢ 1c ∈ NC
42		tcdi 6165	. . . . . . . . . . 11 ⊢ ((a ∈ NC ∧ 1c ∈ NC ) → Tc (a +c 1c) = ( Tc a +c Tc 1c))
43	41, 42	mpan2 652	. . . . . . . . . 10 ⊢ (a ∈ NC → Tc (a +c 1c) = ( Tc a +c Tc 1c))
44		tc1c 6166	. . . . . . . . . . 11 ⊢ Tc 1c = 1c
45	44	addceq2i 4388	. . . . . . . . . 10 ⊢ ( Tc a +c Tc 1c) = ( Tc a +c 1c)
46	43, 45	syl6eq 2401	. . . . . . . . 9 ⊢ (a ∈ NC → Tc (a +c 1c) = ( Tc a +c 1c))
47		tceq 6159	. . . . . . . . 9 ⊢ ( Tc (a +c 1c) = ( Tc a +c 1c) → Tc Tc (a +c 1c) = Tc ( Tc a +c 1c))
48	46, 47	syl 15	. . . . . . . 8 ⊢ (a ∈ NC → Tc Tc (a +c 1c) = Tc ( Tc a +c 1c))
49		tccl 6161	. . . . . . . . . 10 ⊢ (a ∈ NC → Tc a ∈ NC )
50		tcdi 6165	. . . . . . . . . . 11 ⊢ (( Tc a ∈ NC ∧ 1c ∈ NC ) → Tc ( Tc a +c 1c) = ( Tc Tc a +c Tc 1c))
51	41, 50	mpan2 652	. . . . . . . . . 10 ⊢ ( Tc a ∈ NC → Tc ( Tc a +c 1c) = ( Tc Tc a +c Tc 1c))
52	49, 51	syl 15	. . . . . . . . 9 ⊢ (a ∈ NC → Tc ( Tc a +c 1c) = ( Tc Tc a +c Tc 1c))
53	44	addceq2i 4388	. . . . . . . . 9 ⊢ ( Tc Tc a +c Tc 1c) = ( Tc Tc a +c 1c)
54	52, 53	syl6eq 2401	. . . . . . . 8 ⊢ (a ∈ NC → Tc ( Tc a +c 1c) = ( Tc Tc a +c 1c))
55	48, 54	eqtrd 2385	. . . . . . 7 ⊢ (a ∈ NC → Tc Tc (a +c 1c) = ( Tc Tc a +c 1c))
56	40, 55	syl 15	. . . . . 6 ⊢ (a ∈ Nn → Tc Tc (a +c 1c) = ( Tc Tc a +c 1c))
57	56	eleq2d 2420	. . . . 5 ⊢ (a ∈ Nn → ({m ∈ Nn ∣ (1c ≤c m ∧ m ≤c (a +c 1c))} ∈ Tc Tc (a +c 1c) ↔ {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c (a +c 1c))} ∈ ( Tc Tc a +c 1c)))
58	57	adantr 451	. . . 4 ⊢ ((a ∈ Nn ∧ Tc Tc a ∈ Nn ) → ({m ∈ Nn ∣ (1c ≤c m ∧ m ≤c (a +c 1c))} ∈ Tc Tc (a +c 1c) ↔ {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c (a +c 1c))} ∈ ( Tc Tc a +c 1c)))
59	39, 58	sylibrd 225	. . 3 ⊢ ((a ∈ Nn ∧ Tc Tc a ∈ Nn ) → ({m ∈ Nn ∣ (1c ≤c m ∧ m ≤c a)} ∈ Tc Tc a → {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c (a +c 1c))} ∈ Tc Tc (a +c 1c)))
60	38, 59	mpdan 649	. 2 ⊢ (a ∈ Nn → ({m ∈ Nn ∣ (1c ≤c m ∧ m ≤c a)} ∈ Tc Tc a → {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c (a +c 1c))} ∈ Tc Tc (a +c 1c)))
61	1, 8, 15, 22, 29, 35, 60	finds 4412	1 ⊢ (N ∈ Nn → {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c N)} ∈ Tc Tc N)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {crab 2619  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376   class class class wbr 4640   NC cncs 6089   ≤_c clec 6090   T_c ctc 6094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-fix 5741  df-cup 5743  df-disj 5745  df-addcfn 5747  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-clos1 5874  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-lec 6100  df-ltc 6101  df-nc 6102  df-tc 6104  df-tcfn 6108

This theorem is referenced by: (None)