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Theorem nclenn 6249
 Description: A cardinal that is less than or equal to a natural is a natural. Theorem XI.3.3 of [Rosser] p. 391. (Contributed by SF, 19-Mar-2015.)
Assertion
Ref Expression
nclenn ((M NC N Nn Mc N) → M Nn )

Proof of Theorem nclenn
Dummy variables m n p q x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nclennlem1 6248 . . . . 5 {x n NC (nc xn Nn )} V
2 breq2 4643 . . . . . . 7 (x = 0c → (nc xnc 0c))
32imbi1d 308 . . . . . 6 (x = 0c → ((nc xn Nn ) ↔ (nc 0cn Nn )))
43ralbidv 2634 . . . . 5 (x = 0c → (n NC (nc xn Nn ) ↔ n NC (nc 0cn Nn )))
5 breq2 4643 . . . . . . 7 (x = m → (nc xnc m))
65imbi1d 308 . . . . . 6 (x = m → ((nc xn Nn ) ↔ (nc mn Nn )))
76ralbidv 2634 . . . . 5 (x = m → (n NC (nc xn Nn ) ↔ n NC (nc mn Nn )))
8 breq2 4643 . . . . . . 7 (x = (m +c 1c) → (nc xnc (m +c 1c)))
98imbi1d 308 . . . . . 6 (x = (m +c 1c) → ((nc xn Nn ) ↔ (nc (m +c 1c) → n Nn )))
109ralbidv 2634 . . . . 5 (x = (m +c 1c) → (n NC (nc xn Nn ) ↔ n NC (nc (m +c 1c) → n Nn )))
11 breq2 4643 . . . . . . 7 (x = N → (nc xnc N))
1211imbi1d 308 . . . . . 6 (x = N → ((nc xn Nn ) ↔ (nc Nn Nn )))
1312ralbidv 2634 . . . . 5 (x = N → (n NC (nc xn Nn ) ↔ n NC (nc Nn Nn )))
14 le0nc 6200 . . . . . . 7 (n NC → 0cc n)
15 0cnc 6138 . . . . . . . . . . 11 0c NC
16 sbth 6206 . . . . . . . . . . 11 ((n NC 0c NC ) → ((nc 0c 0cc n) → n = 0c))
1715, 16mpan2 652 . . . . . . . . . 10 (n NC → ((nc 0c 0cc n) → n = 0c))
1817imp 418 . . . . . . . . 9 ((n NC (nc 0c 0cc n)) → n = 0c)
19 peano1 4402 . . . . . . . . 9 0c Nn
2018, 19syl6eqel 2441 . . . . . . . 8 ((n NC (nc 0c 0cc n)) → n Nn )
2120ex 423 . . . . . . 7 (n NC → ((nc 0c 0cc n) → n Nn ))
2214, 21mpan2d 655 . . . . . 6 (n NC → (nc 0cn Nn ))
2322rgen 2679 . . . . 5 n NC (nc 0cn Nn )
24 peano2 4403 . . . . . . . . . . . 12 (m Nn → (m +c 1c) Nn )
25 nnnc 6146 . . . . . . . . . . . 12 ((m +c 1c) Nn → (m +c 1c) NC )
2624, 25syl 15 . . . . . . . . . . 11 (m Nn → (m +c 1c) NC )
27 dflec2 6210 . . . . . . . . . . 11 ((n NC (m +c 1c) NC ) → (nc (m +c 1c) ↔ p NC (m +c 1c) = (n +c p)))
2826, 27sylan2 460 . . . . . . . . . 10 ((n NC m Nn ) → (nc (m +c 1c) ↔ p NC (m +c 1c) = (n +c p)))
2928ancoms 439 . . . . . . . . 9 ((m Nn n NC ) → (nc (m +c 1c) ↔ p NC (m +c 1c) = (n +c p)))
30293adant3 975 . . . . . . . 8 ((m Nn n NC (nc mn Nn )) → (nc (m +c 1c) ↔ p NC (m +c 1c) = (n +c p)))
31 nc0suc 6217 . . . . . . . . . 10 (p NC → (p = 0c q NC p = (q +c 1c)))
32 addceq2 4384 . . . . . . . . . . . . . . . . . 18 (p = 0c → (n +c p) = (n +c 0c))
33 addcid1 4405 . . . . . . . . . . . . . . . . . 18 (n +c 0c) = n
3432, 33syl6eq 2401 . . . . . . . . . . . . . . . . 17 (p = 0c → (n +c p) = n)
3534eqeq2d 2364 . . . . . . . . . . . . . . . 16 (p = 0c → ((m +c 1c) = (n +c p) ↔ (m +c 1c) = n))
3635biimpa 470 . . . . . . . . . . . . . . 15 ((p = 0c (m +c 1c) = (n +c p)) → (m +c 1c) = n)
37 eleq1 2413 . . . . . . . . . . . . . . . 16 ((m +c 1c) = n → ((m +c 1c) Nnn Nn ))
3837biimpcd 215 . . . . . . . . . . . . . . 15 ((m +c 1c) Nn → ((m +c 1c) = nn Nn ))
3936, 38syl5 28 . . . . . . . . . . . . . 14 ((m +c 1c) Nn → ((p = 0c (m +c 1c) = (n +c p)) → n Nn ))
4039exp3a 425 . . . . . . . . . . . . 13 ((m +c 1c) Nn → (p = 0c → ((m +c 1c) = (n +c p) → n Nn )))
4124, 40syl 15 . . . . . . . . . . . 12 (m Nn → (p = 0c → ((m +c 1c) = (n +c p) → n Nn )))
42413ad2ant1 976 . . . . . . . . . . 11 ((m Nn n NC (nc mn Nn )) → (p = 0c → ((m +c 1c) = (n +c p) → n Nn )))
43 addceq2 4384 . . . . . . . . . . . . . . . . 17 (p = (q +c 1c) → (n +c p) = (n +c (q +c 1c)))
44 addcass 4415 . . . . . . . . . . . . . . . . 17 ((n +c q) +c 1c) = (n +c (q +c 1c))
4543, 44syl6eqr 2403 . . . . . . . . . . . . . . . 16 (p = (q +c 1c) → (n +c p) = ((n +c q) +c 1c))
4645eqeq2d 2364 . . . . . . . . . . . . . . 15 (p = (q +c 1c) → ((m +c 1c) = (n +c p) ↔ (m +c 1c) = ((n +c q) +c 1c)))
4746biimpa 470 . . . . . . . . . . . . . 14 ((p = (q +c 1c) (m +c 1c) = (n +c p)) → (m +c 1c) = ((n +c q) +c 1c))
48 nnnc 6146 . . . . . . . . . . . . . . . . . 18 (m Nnm NC )
49483ad2ant1 976 . . . . . . . . . . . . . . . . 17 ((m Nn n NC (nc mn Nn )) → m NC )
5049adantr 451 . . . . . . . . . . . . . . . 16 (((m Nn n NC (nc mn Nn )) q NC ) → m NC )
51 ncaddccl 6144 . . . . . . . . . . . . . . . . 17 ((n NC q NC ) → (n +c q) NC )
52513ad2antl2 1118 . . . . . . . . . . . . . . . 16 (((m Nn n NC (nc mn Nn )) q NC ) → (n +c q) NC )
53 peano4nc 6150 . . . . . . . . . . . . . . . 16 ((m NC (n +c q) NC ) → ((m +c 1c) = ((n +c q) +c 1c) ↔ m = (n +c q)))
5450, 52, 53syl2anc 642 . . . . . . . . . . . . . . 15 (((m Nn n NC (nc mn Nn )) q NC ) → ((m +c 1c) = ((n +c q) +c 1c) ↔ m = (n +c q)))
55 addlecncs 6209 . . . . . . . . . . . . . . . . . . . . . . 23 ((n NC q NC ) → nc (n +c q))
56 breq2 4643 . . . . . . . . . . . . . . . . . . . . . . 23 (m = (n +c q) → (nc mnc (n +c q)))
5755, 56syl5ibrcom 213 . . . . . . . . . . . . . . . . . . . . . 22 ((n NC q NC ) → (m = (n +c q) → nc m))
5857ex 423 . . . . . . . . . . . . . . . . . . . . 21 (n NC → (q NC → (m = (n +c q) → nc m)))
5958com23 72 . . . . . . . . . . . . . . . . . . . 20 (n NC → (m = (n +c q) → (q NCnc m)))
6059adantl 452 . . . . . . . . . . . . . . . . . . 19 ((m Nn n NC ) → (m = (n +c q) → (q NCnc m)))
61 pm2.27 35 . . . . . . . . . . . . . . . . . . 19 (nc m → ((nc mn Nn ) → n Nn ))
6260, 61syl8 65 . . . . . . . . . . . . . . . . . 18 ((m Nn n NC ) → (m = (n +c q) → (q NC → ((nc mn Nn ) → n Nn ))))
6362com24 81 . . . . . . . . . . . . . . . . 17 ((m Nn n NC ) → ((nc mn Nn ) → (q NC → (m = (n +c q) → n Nn ))))
64633impia 1148 . . . . . . . . . . . . . . . 16 ((m Nn n NC (nc mn Nn )) → (q NC → (m = (n +c q) → n Nn )))
6564imp 418 . . . . . . . . . . . . . . 15 (((m Nn n NC (nc mn Nn )) q NC ) → (m = (n +c q) → n Nn ))
6654, 65sylbid 206 . . . . . . . . . . . . . 14 (((m Nn n NC (nc mn Nn )) q NC ) → ((m +c 1c) = ((n +c q) +c 1c) → n Nn ))
6747, 66syl5 28 . . . . . . . . . . . . 13 (((m Nn n NC (nc mn Nn )) q NC ) → ((p = (q +c 1c) (m +c 1c) = (n +c p)) → n Nn ))
6867exp3a 425 . . . . . . . . . . . 12 (((m Nn n NC (nc mn Nn )) q NC ) → (p = (q +c 1c) → ((m +c 1c) = (n +c p) → n Nn )))
6968rexlimdva 2738 . . . . . . . . . . 11 ((m Nn n NC (nc mn Nn )) → (q NC p = (q +c 1c) → ((m +c 1c) = (n +c p) → n Nn )))
7042, 69jaod 369 . . . . . . . . . 10 ((m Nn n NC (nc mn Nn )) → ((p = 0c q NC p = (q +c 1c)) → ((m +c 1c) = (n +c p) → n Nn )))
7131, 70syl5 28 . . . . . . . . 9 ((m Nn n NC (nc mn Nn )) → (p NC → ((m +c 1c) = (n +c p) → n Nn )))
7271rexlimdv 2737 . . . . . . . 8 ((m Nn n NC (nc mn Nn )) → (p NC (m +c 1c) = (n +c p) → n Nn ))
7330, 72sylbid 206 . . . . . . 7 ((m Nn n NC (nc mn Nn )) → (nc (m +c 1c) → n Nn ))
74733expia 1153 . . . . . 6 ((m Nn n NC ) → ((nc mn Nn ) → (nc (m +c 1c) → n Nn )))
7574ralimdva 2692 . . . . 5 (m Nn → (n NC (nc mn Nn ) → n NC (nc (m +c 1c) → n Nn )))
761, 4, 7, 10, 13, 23, 75finds 4411 . . . 4 (N Nnn NC (nc Nn Nn ))
77 breq1 4642 . . . . . 6 (n = M → (nc NMc N))
78 eleq1 2413 . . . . . 6 (n = M → (n NnM Nn ))
7977, 78imbi12d 311 . . . . 5 (n = M → ((nc Nn Nn ) ↔ (Mc NM Nn )))
8079rspccv 2952 . . . 4 (n NC (nc Nn Nn ) → (M NC → (Mc NM Nn )))
8176, 80syl 15 . . 3 (N Nn → (M NC → (Mc NM Nn )))
8281com12 27 . 2 (M NC → (N Nn → (Mc NM Nn )))
83823imp 1145 1 ((M NC N Nn Mc N) → M Nn )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615  1cc1c 4134   Nn cnnc 4373  0cc0c 4374   +c cplc 4375   class class class wbr 4639   NC cncs 6088   ≤c clec 6089 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-txp 5736  df-fix 5740  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-clos1 5873  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-lec 6099  df-nc 6101 This theorem is referenced by:  nchoicelem17  6305
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