New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  elsuc GIF version

Theorem elsuc 4413
 Description: Membership in a successor. Theorem X.1.16 of [Rosser] p. 279. (Contributed by SF, 16-Jan-2015.)
Assertion
Ref Expression
elsuc (A (M +c 1c) ↔ b M x bA = (b ∪ {x}))
Distinct variable groups:   A,b,x   M,b
Allowed substitution hint:   M(x)

Proof of Theorem elsuc
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eladdc 4398 . 2 (A (M +c 1c) ↔ b M y 1c ((by) = A = (by)))
2 snex 4111 . . . . . . 7 {x} V
3 ineq2 3451 . . . . . . . . 9 (y = {x} → (by) = (b ∩ {x}))
43eqeq1d 2361 . . . . . . . 8 (y = {x} → ((by) = ↔ (b ∩ {x}) = ))
5 uneq2 3412 . . . . . . . . 9 (y = {x} → (by) = (b ∪ {x}))
65eqeq2d 2364 . . . . . . . 8 (y = {x} → (A = (by) ↔ A = (b ∪ {x})))
74, 6anbi12d 691 . . . . . . 7 (y = {x} → (((by) = A = (by)) ↔ ((b ∩ {x}) = A = (b ∪ {x}))))
82, 7ceqsexv 2894 . . . . . 6 (y(y = {x} ((by) = A = (by))) ↔ ((b ∩ {x}) = A = (b ∪ {x})))
9 disjsn 3786 . . . . . . . 8 ((b ∩ {x}) = ↔ ¬ x b)
10 vex 2862 . . . . . . . . 9 x V
1110elcompl 3225 . . . . . . . 8 (x b ↔ ¬ x b)
129, 11bitr4i 243 . . . . . . 7 ((b ∩ {x}) = x b)
1312anbi1i 676 . . . . . 6 (((b ∩ {x}) = A = (b ∪ {x})) ↔ (x b A = (b ∪ {x})))
148, 13bitri 240 . . . . 5 (y(y = {x} ((by) = A = (by))) ↔ (x b A = (b ∪ {x})))
1514exbii 1582 . . . 4 (xy(y = {x} ((by) = A = (by))) ↔ x(x b A = (b ∪ {x})))
16 df-rex 2620 . . . . 5 (y 1c ((by) = A = (by)) ↔ y(y 1c ((by) = A = (by))))
17 el1c 4139 . . . . . . . . 9 (y 1cx y = {x})
1817anbi1i 676 . . . . . . . 8 ((y 1c ((by) = A = (by))) ↔ (x y = {x} ((by) = A = (by))))
19 19.41v 1901 . . . . . . . 8 (x(y = {x} ((by) = A = (by))) ↔ (x y = {x} ((by) = A = (by))))
2018, 19bitr4i 243 . . . . . . 7 ((y 1c ((by) = A = (by))) ↔ x(y = {x} ((by) = A = (by))))
2120exbii 1582 . . . . . 6 (y(y 1c ((by) = A = (by))) ↔ yx(y = {x} ((by) = A = (by))))
22 excom 1741 . . . . . 6 (yx(y = {x} ((by) = A = (by))) ↔ xy(y = {x} ((by) = A = (by))))
2321, 22bitri 240 . . . . 5 (y(y 1c ((by) = A = (by))) ↔ xy(y = {x} ((by) = A = (by))))
2416, 23bitri 240 . . . 4 (y 1c ((by) = A = (by)) ↔ xy(y = {x} ((by) = A = (by))))
25 df-rex 2620 . . . 4 (x bA = (b ∪ {x}) ↔ x(x b A = (b ∪ {x})))
2615, 24, 253bitr4i 268 . . 3 (y 1c ((by) = A = (by)) ↔ x bA = (b ∪ {x}))
2726rexbii 2639 . 2 (b M y 1c ((by) = A = (by)) ↔ b M x bA = (b ∪ {x}))
281, 27bitri 240 1 (A (M +c 1c) ↔ b M x bA = (b ∪ {x}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∼ ccompl 3205   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  {csn 3737  1cc1c 4134   +c cplc 4375 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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-1c 4136  df-addc 4378 This theorem is referenced by:  elsuci  4414  nnsucelr  4428  nndisjeq  4429  prepeano4  4451  ncfinraise  4481  ncfinlower  4483  tfinsuc  4498  oddfinex  4504  nnadjoin  4520  nnpweq  4523  sfindbl  4530  tfinnn  4534  peano4nc  6150  el2c  6191  nmembers1lem3  6270
 Copyright terms: Public domain W3C validator