Theorem nmembers1lem2 6270

Description: Lemma for nmembers1 6272. The set of all elements between one and zero is empty. (Contributed by Scott Fenton, 1-Aug-2019.)

Assertion

Ref	Expression
nmembers1lem2	⊢ {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c 0c)} ∈ 0c

Proof of Theorem nmembers1lem2

Step	Hyp	Ref	Expression
1		0lt1c 6259	. . . . . . . 8 ⊢ 0c <c 1c
2		0cnc 6139	. . . . . . . . 9 ⊢ 0c ∈ NC
3		1cnc 6140	. . . . . . . . 9 ⊢ 1c ∈ NC
4		ltlenlec 6208	. . . . . . . . 9 ⊢ ((0c ∈ NC ∧ 1c ∈ NC ) → (0c <c 1c ↔ (0c ≤c 1c ∧ ¬ 1c ≤c 0c)))
5	2, 3, 4	mp2an 653	. . . . . . . 8 ⊢ (0c <c 1c ↔ (0c ≤c 1c ∧ ¬ 1c ≤c 0c))
6	1, 5	mpbi 199	. . . . . . 7 ⊢ (0c ≤c 1c ∧ ¬ 1c ≤c 0c)
7	6	simpri 448	. . . . . 6 ⊢ ¬ 1c ≤c 0c
8		nnnc 6147	. . . . . . . . 9 ⊢ (m ∈ Nn → m ∈ NC )
9		lectr 6212	. . . . . . . . . 10 ⊢ ((1c ∈ NC ∧ m ∈ NC ∧ 0c ∈ NC ) → ((1c ≤c m ∧ m ≤c 0c) → 1c ≤c 0c))
10	3, 2, 9	mp3an13 1268	. . . . . . . . 9 ⊢ (m ∈ NC → ((1c ≤c m ∧ m ≤c 0c) → 1c ≤c 0c))
11	8, 10	syl 15	. . . . . . . 8 ⊢ (m ∈ Nn → ((1c ≤c m ∧ m ≤c 0c) → 1c ≤c 0c))
12	11	exp3a 425	. . . . . . 7 ⊢ (m ∈ Nn → (1c ≤c m → (m ≤c 0c → 1c ≤c 0c)))
13	12	imp 418	. . . . . 6 ⊢ ((m ∈ Nn ∧ 1c ≤c m) → (m ≤c 0c → 1c ≤c 0c))
14	7, 13	mtoi 169	. . . . 5 ⊢ ((m ∈ Nn ∧ 1c ≤c m) → ¬ m ≤c 0c)
15	14	ex 423	. . . 4 ⊢ (m ∈ Nn → (1c ≤c m → ¬ m ≤c 0c))
16		imnan 411	. . . 4 ⊢ ((1c ≤c m → ¬ m ≤c 0c) ↔ ¬ (1c ≤c m ∧ m ≤c 0c))
17	15, 16	sylib 188	. . 3 ⊢ (m ∈ Nn → ¬ (1c ≤c m ∧ m ≤c 0c))
18	17	rgen 2680	. 2 ⊢ ∀m ∈ Nn ¬ (1c ≤c m ∧ m ≤c 0c)
19		el0c 4422	. . 3 ⊢ ({m ∈ Nn ∣ (1c ≤c m ∧ m ≤c 0c)} ∈ 0c ↔ {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c 0c)} = ∅)
20		rabeq0 3573	. . 3 ⊢ ({m ∈ Nn ∣ (1c ≤c m ∧ m ≤c 0c)} = ∅ ↔ ∀m ∈ Nn ¬ (1c ≤c m ∧ m ≤c 0c))
21	19, 20	bitri 240	. 2 ⊢ ({m ∈ Nn ∣ (1c ≤c m ∧ m ≤c 0c)} ∈ 0c ↔ ∀m ∈ Nn ¬ (1c ≤c m ∧ m ≤c 0c))
22	18, 21	mpbir 200	1 ⊢ {m ∈ Nn ∣ (1c ≤c m ∧ m ≤c 0c)} ∈ 0c

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615  {crab 2619  ∅c0 3551  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   class class class wbr 4640   NC cncs 6089   ≤_c clec 6090   <_c cltc 6091

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-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-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-txp 5737  df-fix 5741  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  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

This theorem is referenced by:  nmembers1  6272