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Theorem lectr 6211
 Description: Cardinal less than or equal is transitive. (Contributed by SF, 12-Mar-2015.)
Assertion
Ref Expression
lectr ((A NC B NC C NC ) → ((Ac B Bc C) → Ac C))

Proof of Theorem lectr
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dflec2 6210 . . . . 5 ((A NC B NC ) → (Ac Bx NC B = (A +c x)))
213adant3 975 . . . 4 ((A NC B NC C NC ) → (Ac Bx NC B = (A +c x)))
3 dflec2 6210 . . . . 5 ((B NC C NC ) → (Bc Cy NC C = (B +c y)))
433adant1 973 . . . 4 ((A NC B NC C NC ) → (Bc Cy NC C = (B +c y)))
52, 4anbi12d 691 . . 3 ((A NC B NC C NC ) → ((Ac B Bc C) ↔ (x NC B = (A +c x) y NC C = (B +c y))))
6 reeanv 2778 . . 3 (x NC y NC (B = (A +c x) C = (B +c y)) ↔ (x NC B = (A +c x) y NC C = (B +c y)))
75, 6syl6bbr 254 . 2 ((A NC B NC C NC ) → ((Ac B Bc C) ↔ x NC y NC (B = (A +c x) C = (B +c y))))
8 addceq1 4383 . . . . . . 7 (B = (A +c x) → (B +c y) = ((A +c x) +c y))
9 addcass 4415 . . . . . . 7 ((A +c x) +c y) = (A +c (x +c y))
108, 9syl6eq 2401 . . . . . 6 (B = (A +c x) → (B +c y) = (A +c (x +c y)))
1110eqeq2d 2364 . . . . 5 (B = (A +c x) → (C = (B +c y) ↔ C = (A +c (x +c y))))
1211biimpa 470 . . . 4 ((B = (A +c x) C = (B +c y)) → C = (A +c (x +c y)))
13 simp1 955 . . . . . 6 ((A NC B NC C NC ) → A NC )
14 ncaddccl 6144 . . . . . 6 ((x NC y NC ) → (x +c y) NC )
15 addlecncs 6209 . . . . . 6 ((A NC (x +c y) NC ) → Ac (A +c (x +c y)))
1613, 14, 15syl2an 463 . . . . 5 (((A NC B NC C NC ) (x NC y NC )) → Ac (A +c (x +c y)))
17 breq2 4643 . . . . 5 (C = (A +c (x +c y)) → (Ac CAc (A +c (x +c y))))
1816, 17syl5ibrcom 213 . . . 4 (((A NC B NC C NC ) (x NC y NC )) → (C = (A +c (x +c y)) → Ac C))
1912, 18syl5 28 . . 3 (((A NC B NC C NC ) (x NC y NC )) → ((B = (A +c x) C = (B +c y)) → Ac C))
2019rexlimdvva 2745 . 2 ((A NC B NC C NC ) → (x NC y NC (B = (A +c x) C = (B +c y)) → Ac C))
217, 20sylbid 206 1 ((A NC B NC C NC ) → ((Ac B Bc C) → Ac C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   +c cplc 4375   class class class wbr 4639   NC cncs 6088   ≤c clec 6089 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  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:  leltctr  6212  lecponc  6213  leconnnc  6218  ncslesuc  6267  nmembers1lem2  6269  nchoicelem4  6292
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