Theorem elsuc 4414

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 4399	. 2 |- (A e. (M +o 1c) <-> E.b e. M E.y e. 1c ((b i^i y) = (/) /\ A = (b u. y)))
2		snex 4112	. . . . . . 7 |- {x} e. _V
3		ineq2 3452	. . . . . . . . 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 3413	. . . . . . . . 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 2895	. . . . . 6 |- (E.y(y = {x} /\ ((b i^i y) = (/) /\ A = (b u. y))) <-> ((b i^i {x}) = (/) /\ A = (b u. {x})))
9		disjsn 3787	. . . . . . . 8 |- ((b i^i {x}) = (/) <-> -. x e. b)
10		vex 2863	. . . . . . . . 9 |- x e. _V
11	10	elcompl 3226	. . . . . . . 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 2621	. . . . 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 4140	. . . . . . . . 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 2621	. . . 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 2640	. 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 2616   ∼ ccompl 3206   u. cun 3208   i^i cin 3209  (/)c0 3551  {csn 3738  1_cc1c 4135   +o cplc 4376

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-1c 4137  df-addc 4379

This theorem is referenced by:  elsuci  4415  nnsucelr  4429  nndisjeq  4430  prepeano4  4452  ncfinraise  4482  ncfinlower  4484  tfinsuc  4499  oddfinex  4505  nnadjoin  4521  nnpweq  4524  sfindbl  4531  tfinnn  4535  peano4nc  6151  el2c  6192  nmembers1lem3  6271