Theorem ncslesuc 6268

Description: Relationship between successor and cardinal less than or equal. (Contributed by Scott Fenton, 3-Aug-2019.)

Assertion

Ref	Expression
ncslesuc	⊢ ((M ∈ NC ∧ N ∈ NC ) → (M ≤c (N +c 1c) ↔ (M ≤c N ∨ M = (N +c 1c))))

Proof of Theorem ncslesuc

Dummy variables p q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2nc 6146	. . . 4 ⊢ (N ∈ NC → (N +c 1c) ∈ NC )
2		dflec2 6211	. . . 4 ⊢ ((M ∈ NC ∧ (N +c 1c) ∈ NC ) → (M ≤c (N +c 1c) ↔ ∃p ∈ NC (N +c 1c) = (M +c p)))
3	1, 2	sylan2 460	. . 3 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (M ≤c (N +c 1c) ↔ ∃p ∈ NC (N +c 1c) = (M +c p)))
4		nc0suc 6218	. . . . 5 ⊢ (p ∈ NC → (p = 0c ∨ ∃q ∈ NC p = (q +c 1c)))
5		addceq2 4385	. . . . . . . . . 10 ⊢ (p = 0c → (M +c p) = (M +c 0c))
6		addcid1 4406	. . . . . . . . . 10 ⊢ (M +c 0c) = M
7	5, 6	syl6eq 2401	. . . . . . . . 9 ⊢ (p = 0c → (M +c p) = M)
8	7	eqeq2d 2364	. . . . . . . 8 ⊢ (p = 0c → ((N +c 1c) = (M +c p) ↔ (N +c 1c) = M))
9		olc 373	. . . . . . . . 9 ⊢ (M = (N +c 1c) → (M ≤c N ∨ M = (N +c 1c)))
10	9	eqcoms 2356	. . . . . . . 8 ⊢ ((N +c 1c) = M → (M ≤c N ∨ M = (N +c 1c)))
11	8, 10	syl6bi 219	. . . . . . 7 ⊢ (p = 0c → ((N +c 1c) = (M +c p) → (M ≤c N ∨ M = (N +c 1c))))
12	11	a1i 10	. . . . . 6 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (p = 0c → ((N +c 1c) = (M +c p) → (M ≤c N ∨ M = (N +c 1c)))))
13		addceq2 4385	. . . . . . . . . . . 12 ⊢ (p = (q +c 1c) → (M +c p) = (M +c (q +c 1c)))
14		addcass 4416	. . . . . . . . . . . 12 ⊢ ((M +c q) +c 1c) = (M +c (q +c 1c))
15	13, 14	syl6eqr 2403	. . . . . . . . . . 11 ⊢ (p = (q +c 1c) → (M +c p) = ((M +c q) +c 1c))
16	15	eqeq2d 2364	. . . . . . . . . 10 ⊢ (p = (q +c 1c) → ((N +c 1c) = (M +c p) ↔ (N +c 1c) = ((M +c q) +c 1c)))
17	16	biimpa 470	. . . . . . . . 9 ⊢ ((p = (q +c 1c) ∧ (N +c 1c) = (M +c p)) → (N +c 1c) = ((M +c q) +c 1c))
18		simplr 731	. . . . . . . . . . 11 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ q ∈ NC ) → N ∈ NC )
19		ncaddccl 6145	. . . . . . . . . . . 12 ⊢ ((M ∈ NC ∧ q ∈ NC ) → (M +c q) ∈ NC )
20	19	adantlr 695	. . . . . . . . . . 11 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ q ∈ NC ) → (M +c q) ∈ NC )
21		peano4nc 6151	. . . . . . . . . . 11 ⊢ ((N ∈ NC ∧ (M +c q) ∈ NC ) → ((N +c 1c) = ((M +c q) +c 1c) ↔ N = (M +c q)))
22	18, 20, 21	syl2anc 642	. . . . . . . . . 10 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ q ∈ NC ) → ((N +c 1c) = ((M +c q) +c 1c) ↔ N = (M +c q)))
23		addlecncs 6210	. . . . . . . . . . . . 13 ⊢ ((M ∈ NC ∧ q ∈ NC ) → M ≤c (M +c q))
24	23	adantlr 695	. . . . . . . . . . . 12 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ q ∈ NC ) → M ≤c (M +c q))
25		breq2 4644	. . . . . . . . . . . 12 ⊢ (N = (M +c q) → (M ≤c N ↔ M ≤c (M +c q)))
26	24, 25	syl5ibrcom 213	. . . . . . . . . . 11 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ q ∈ NC ) → (N = (M +c q) → M ≤c N))
27		orc 374	. . . . . . . . . . 11 ⊢ (M ≤c N → (M ≤c N ∨ M = (N +c 1c)))
28	26, 27	syl6 29	. . . . . . . . . 10 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ q ∈ NC ) → (N = (M +c q) → (M ≤c N ∨ M = (N +c 1c))))
29	22, 28	sylbid 206	. . . . . . . . 9 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ q ∈ NC ) → ((N +c 1c) = ((M +c q) +c 1c) → (M ≤c N ∨ M = (N +c 1c))))
30	17, 29	syl5 28	. . . . . . . 8 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ q ∈ NC ) → ((p = (q +c 1c) ∧ (N +c 1c) = (M +c p)) → (M ≤c N ∨ M = (N +c 1c))))
31	30	exp3a 425	. . . . . . 7 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ q ∈ NC ) → (p = (q +c 1c) → ((N +c 1c) = (M +c p) → (M ≤c N ∨ M = (N +c 1c)))))
32	31	rexlimdva 2739	. . . . . 6 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (∃q ∈ NC p = (q +c 1c) → ((N +c 1c) = (M +c p) → (M ≤c N ∨ M = (N +c 1c)))))
33	12, 32	jaod 369	. . . . 5 ⊢ ((M ∈ NC ∧ N ∈ NC ) → ((p = 0c ∨ ∃q ∈ NC p = (q +c 1c)) → ((N +c 1c) = (M +c p) → (M ≤c N ∨ M = (N +c 1c)))))
34	4, 33	syl5 28	. . . 4 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (p ∈ NC → ((N +c 1c) = (M +c p) → (M ≤c N ∨ M = (N +c 1c)))))
35	34	rexlimdv 2738	. . 3 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (∃p ∈ NC (N +c 1c) = (M +c p) → (M ≤c N ∨ M = (N +c 1c))))
36	3, 35	sylbid 206	. 2 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (M ≤c (N +c 1c) → (M ≤c N ∨ M = (N +c 1c))))
37		1cnc 6140	. . . . . 6 ⊢ 1c ∈ NC
38		addlecncs 6210	. . . . . 6 ⊢ ((N ∈ NC ∧ 1c ∈ NC ) → N ≤c (N +c 1c))
39	37, 38	mpan2 652	. . . . 5 ⊢ (N ∈ NC → N ≤c (N +c 1c))
40	39	adantl 452	. . . 4 ⊢ ((M ∈ NC ∧ N ∈ NC ) → N ≤c (N +c 1c))
41	1	adantl 452	. . . . 5 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (N +c 1c) ∈ NC )
42		lectr 6212	. . . . 5 ⊢ ((M ∈ NC ∧ N ∈ NC ∧ (N +c 1c) ∈ NC ) → ((M ≤c N ∧ N ≤c (N +c 1c)) → M ≤c (N +c 1c)))
43	41, 42	mpd3an3 1278	. . . 4 ⊢ ((M ∈ NC ∧ N ∈ NC ) → ((M ≤c N ∧ N ≤c (N +c 1c)) → M ≤c (N +c 1c)))
44	40, 43	mpan2d 655	. . 3 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (M ≤c N → M ≤c (N +c 1c)))
45		nclecid 6198	. . . . . 6 ⊢ ((N +c 1c) ∈ NC → (N +c 1c) ≤c (N +c 1c))
46	1, 45	syl 15	. . . . 5 ⊢ (N ∈ NC → (N +c 1c) ≤c (N +c 1c))
47	46	adantl 452	. . . 4 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (N +c 1c) ≤c (N +c 1c))
48		breq1 4643	. . . 4 ⊢ (M = (N +c 1c) → (M ≤c (N +c 1c) ↔ (N +c 1c) ≤c (N +c 1c)))
49	47, 48	syl5ibrcom 213	. . 3 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (M = (N +c 1c) → M ≤c (N +c 1c)))
50	44, 49	jaod 369	. 2 ⊢ ((M ∈ NC ∧ N ∈ NC ) → ((M ≤c N ∨ M = (N +c 1c)) → M ≤c (N +c 1c)))
51	36, 50	impbid 183	1 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (M ≤c (N +c 1c) ↔ (M ≤c N ∨ M = (N +c 1c))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  1_cc1c 4135  0_cc0c 4375   +_c cplc 4376   class class class wbr 4640   NC cncs 6089   ≤_c clec 6090

This proof depends on 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 proof 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-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  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-nc 6102

This theorem is used by:  nmembers1lem3  6271