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 e. (M +o 1c) <-> E.b e. M E.x e. ∼ bA = (b u. {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.

Step	Hyp	Ref	Expression
1		eladdc 4398	. 2 |- (A e. (M +o 1c) <-> E.b e. M E.y e. 1c ((b i^i y) = (/) /\ A = (b u. y)))
2		snex 4111	. . . . . . 7 |- {x} e. _V
3		ineq2 3451	. . . . . . . . 9 |- (y = {x} -> (b i^i y) = (b i^i {x}))
4	3	eqeq1d 2361	. . . . . . . 8 |- (y = {x} -> ((b i^i y) = (/) <-> (b i^i {x}) = (/)))
5		uneq2 3412	. . . . . . . . 9 |- (y = {x} -> (b u. y) = (b u. {x}))
6	5	eqeq2d 2364	. . . . . . . 8 |- (y = {x} -> (A = (b u. y) <-> A = (b u. {x})))
7	4, 6	anbi12d 691	. . . . . . 7 |- (y = {x} -> (((b i^i y) = (/) /\ A = (b u. y)) <-> ((b i^i {x}) = (/) /\ A = (b u. {x}))))
8	2, 7	ceqsexv 2894	. . . . . 6 |- (E.y(y = {x} /\ ((b i^i y) = (/) /\ A = (b u. y))) <-> ((b i^i {x}) = (/) /\ A = (b u. {x})))
9		disjsn 3786	. . . . . . . 8 |- ((b i^i {x}) = (/) <-> -. x e. b)
10		vex 2862	. . . . . . . . 9 |- x e. _V
11	10	elcompl 3225	. . . . . . . 8 |- (x e. ∼ b <-> -. x e. b)
12	9, 11	bitr4i 243	. . . . . . 7 |- ((b i^i {x}) = (/) <-> x e. ∼ b)
13	12	anbi1i 676	. . . . . 6 |- (((b i^i {x}) = (/) /\ A = (b u. {x})) <-> (x e. ∼ b /\ A = (b u. {x})))
14	8, 13	bitri 240	. . . . 5 |- (E.y(y = {x} /\ ((b i^i y) = (/) /\ A = (b u. y))) <-> (x e. ∼ b /\ A = (b u. {x})))
15	14	exbii 1582	. . . 4 |- (E.xE.y(y = {x} /\ ((b i^i y) = (/) /\ A = (b u. y))) <-> E.x(x e. ∼ b /\ A = (b u. {x})))
16		df-rex 2620	. . . . 5 |- (E.y e. 1c ((b i^i y) = (/) /\ A = (b u. y)) <-> E.y(y e. 1c /\ ((b i^i y) = (/) /\ A = (b u. y))))
17		el1c 4139	. . . . . . . . 9 |- (y e. 1c <-> E.x y = {x})
18	17	anbi1i 676	. . . . . . . 8 |- ((y e. 1c /\ ((b i^i y) = (/) /\ A = (b u. y))) <-> (E.x y = {x} /\ ((b i^i y) = (/) /\ A = (b u. y))))
19		19.41v 1901	. . . . . . . 8 |- (E.x(y = {x} /\ ((b i^i y) = (/) /\ A = (b u. y))) <-> (E.x y = {x} /\ ((b i^i y) = (/) /\ A = (b u. y))))
20	18, 19	bitr4i 243	. . . . . . 7 |- ((y e. 1c /\ ((b i^i y) = (/) /\ A = (b u. y))) <-> E.x(y = {x} /\ ((b i^i y) = (/) /\ A = (b u. y))))
21	20	exbii 1582	. . . . . 6 |- (E.y(y e. 1c /\ ((b i^i y) = (/) /\ A = (b u. y))) <-> E.yE.x(y = {x} /\ ((b i^i y) = (/) /\ A = (b u. y))))
22		excom 1741	. . . . . 6 |- (E.yE.x(y = {x} /\ ((b i^i y) = (/) /\ A = (b u. y))) <-> E.xE.y(y = {x} /\ ((b i^i y) = (/) /\ A = (b u. y))))
23	21, 22	bitri 240	. . . . 5 |- (E.y(y e. 1c /\ ((b i^i y) = (/) /\ A = (b u. y))) <-> E.xE.y(y = {x} /\ ((b i^i y) = (/) /\ A = (b u. y))))
24	16, 23	bitri 240	. . . 4 |- (E.y e. 1c ((b i^i y) = (/) /\ A = (b u. y)) <-> E.xE.y(y = {x} /\ ((b i^i y) = (/) /\ A = (b u. y))))
25		df-rex 2620	. . . 4 |- (E.x e. ∼ bA = (b u. {x}) <-> E.x(x e. ∼ b /\ A = (b u. {x})))
26	15, 24, 25	3bitr4i 268	. . 3 |- (E.y e. 1c ((b i^i y) = (/) /\ A = (b u. y)) <-> E.x e. ∼ bA = (b u. {x}))
27	26	rexbii 2639	. 2 |- (E.b e. M E.y e. 1c ((b i^i y) = (/) /\ A = (b u. y)) <-> E.b e. M E.x e. ∼ bA = (b u. {x}))
28	1, 27	bitri 240	1 |- (A e. (M +o 1c) <-> E.b e. M E.x e. ∼ bA = (b u. {x}))

Syntax hints:  -. wn 3   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2615   ∼ ccompl 3205   u. cun 3207   i^i cin 3208  (/)c0 3550  {csn 3737  1_cc1c 4134   +o 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