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Theorem dfnnc3 5885
 Description: The finite cardinals as expressed via the closure operation. Theorem X.1.3 of [Rosser] p. 276. (Contributed by SF, 12-Feb-2015.)
Assertion
Ref Expression
dfnnc3 Nn = Clos1 ({0c}, (x V (x +c 1c)))

Proof of Theorem dfnnc3
Dummy variables a y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0cex 4392 . . . . . 6 0c V
21snss 3838 . . . . 5 (0c a ↔ {0c} a)
3 dfss2 3262 . . . . . 6 (((x V (x +c 1c)) “ a) az(z ((x V (x +c 1c)) “ a) → z a))
4 ralcom4 2877 . . . . . . 7 (y a z(((x V (x +c 1c)) ‘y) = zz a) ↔ zy a (((x V (x +c 1c)) ‘y) = zz a))
5 eqid 2353 . . . . . . . . . . . . 13 (x V (x +c 1c)) = (x V (x +c 1c))
65fnmpt 5689 . . . . . . . . . . . 12 (x V (x +c 1c) V → (x V (x +c 1c)) Fn V)
7 vex 2862 . . . . . . . . . . . . . 14 x V
8 1cex 4142 . . . . . . . . . . . . . 14 1c V
97, 8addcex 4394 . . . . . . . . . . . . 13 (x +c 1c) V
109a1i 10 . . . . . . . . . . . 12 (x V → (x +c 1c) V)
116, 10mprg 2683 . . . . . . . . . . 11 (x V (x +c 1c)) Fn V
12 ssv 3291 . . . . . . . . . . 11 a V
13 fvelimab 5370 . . . . . . . . . . 11 (((x V (x +c 1c)) Fn V a V) → (z ((x V (x +c 1c)) “ a) ↔ y a ((x V (x +c 1c)) ‘y) = z))
1411, 12, 13mp2an 653 . . . . . . . . . 10 (z ((x V (x +c 1c)) “ a) ↔ y a ((x V (x +c 1c)) ‘y) = z)
1514imbi1i 315 . . . . . . . . 9 ((z ((x V (x +c 1c)) “ a) → z a) ↔ (y a ((x V (x +c 1c)) ‘y) = zz a))
16 r19.23v 2730 . . . . . . . . 9 (y a (((x V (x +c 1c)) ‘y) = zz a) ↔ (y a ((x V (x +c 1c)) ‘y) = zz a))
1715, 16bitr4i 243 . . . . . . . 8 ((z ((x V (x +c 1c)) “ a) → z a) ↔ y a (((x V (x +c 1c)) ‘y) = zz a))
1817albii 1566 . . . . . . 7 (z(z ((x V (x +c 1c)) “ a) → z a) ↔ zy a (((x V (x +c 1c)) ‘y) = zz a))
194, 18bitr4i 243 . . . . . 6 (y a z(((x V (x +c 1c)) ‘y) = zz a) ↔ z(z ((x V (x +c 1c)) “ a) → z a))
20 vex 2862 . . . . . . . . . . . . 13 y V
21 addceq1 4383 . . . . . . . . . . . . . 14 (x = y → (x +c 1c) = (y +c 1c))
2220, 8addcex 4394 . . . . . . . . . . . . . 14 (y +c 1c) V
2321, 5, 22fvmpt 5700 . . . . . . . . . . . . 13 (y V → ((x V (x +c 1c)) ‘y) = (y +c 1c))
2420, 23ax-mp 5 . . . . . . . . . . . 12 ((x V (x +c 1c)) ‘y) = (y +c 1c)
2524eqeq1i 2360 . . . . . . . . . . 11 (((x V (x +c 1c)) ‘y) = z ↔ (y +c 1c) = z)
26 eqcom 2355 . . . . . . . . . . 11 ((y +c 1c) = zz = (y +c 1c))
2725, 26bitri 240 . . . . . . . . . 10 (((x V (x +c 1c)) ‘y) = zz = (y +c 1c))
2827imbi1i 315 . . . . . . . . 9 ((((x V (x +c 1c)) ‘y) = zz a) ↔ (z = (y +c 1c) → z a))
2928albii 1566 . . . . . . . 8 (z(((x V (x +c 1c)) ‘y) = zz a) ↔ z(z = (y +c 1c) → z a))
30 eleq1 2413 . . . . . . . . 9 (z = (y +c 1c) → (z a ↔ (y +c 1c) a))
3122, 30ceqsalv 2885 . . . . . . . 8 (z(z = (y +c 1c) → z a) ↔ (y +c 1c) a)
3229, 31bitri 240 . . . . . . 7 (z(((x V (x +c 1c)) ‘y) = zz a) ↔ (y +c 1c) a)
3332ralbii 2638 . . . . . 6 (y a z(((x V (x +c 1c)) ‘y) = zz a) ↔ y a (y +c 1c) a)
343, 19, 333bitr2ri 265 . . . . 5 (y a (y +c 1c) a ↔ ((x V (x +c 1c)) “ a) a)
352, 34anbi12i 678 . . . 4 ((0c a y a (y +c 1c) a) ↔ ({0c} a ((x V (x +c 1c)) “ a) a))
3635abbii 2465 . . 3 {a (0c a y a (y +c 1c) a)} = {a ({0c} a ((x V (x +c 1c)) “ a) a)}
3736inteqi 3930 . 2 {a (0c a y a (y +c 1c) a)} = {a ({0c} a ((x V (x +c 1c)) “ a) a)}
38 df-nnc 4379 . 2 Nn = {a (0c a y a (y +c 1c) a)}
39 df-clos1 5873 . 2 Clos1 ({0c}, (x V (x +c 1c))) = {a ({0c} a ((x V (x +c 1c)) “ a) a)}
4037, 38, 393eqtr4i 2383 1 Nn = Clos1 ({0c}, (x V (x +c 1c)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  ∃wrex 2615  Vcvv 2859   ⊆ wss 3257  {csn 3737  ∩cint 3926  1cc1c 4134   Nn cnnc 4373  0cc0c 4374   +c cplc 4375   “ cima 4722   Fn wfn 4776   ‘cfv 4781   ↦ cmpt 5651   Clos1 cclos1 5872 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-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fv 4795  df-mpt 5652  df-clos1 5873 This theorem is referenced by: (None)
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