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 ∈ (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.

Step	Hyp	Ref	Expression
1		eladdc 4399	. 2 ⊢ (A ∈ (M +c 1c) ↔ ∃b ∈ M ∃y ∈ 1c ((b ∩ y) = ∅ ∧ A = (b ∪ y)))
2		snex 4112	. . . . . . 7 ⊢ {x} ∈ V
3		ineq2 3452	. . . . . . . . 9 ⊢ (y = {x} → (b ∩ y) = (b ∩ {x}))
4	3	eqeq1d 2361	. . . . . . . 8 ⊢ (y = {x} → ((b ∩ y) = ∅ ↔ (b ∩ {x}) = ∅))
5		uneq2 3413	. . . . . . . . 9 ⊢ (y = {x} → (b ∪ y) = (b ∪ {x}))
6	5	eqeq2d 2364	. . . . . . . 8 ⊢ (y = {x} → (A = (b ∪ y) ↔ A = (b ∪ {x})))
7	4, 6	anbi12d 691	. . . . . . 7 ⊢ (y = {x} → (((b ∩ y) = ∅ ∧ A = (b ∪ y)) ↔ ((b ∩ {x}) = ∅ ∧ A = (b ∪ {x}))))
8	2, 7	ceqsexv 2895	. . . . . 6 ⊢ (∃y(y = {x} ∧ ((b ∩ y) = ∅ ∧ A = (b ∪ y))) ↔ ((b ∩ {x}) = ∅ ∧ A = (b ∪ {x})))
9		disjsn 3787	. . . . . . . 8 ⊢ ((b ∩ {x}) = ∅ ↔ ¬ x ∈ b)
10		vex 2863	. . . . . . . . 9 ⊢ x ∈ V
11	10	elcompl 3226	. . . . . . . 8 ⊢ (x ∈ ∼ b ↔ ¬ x ∈ b)
12	9, 11	bitr4i 243	. . . . . . 7 ⊢ ((b ∩ {x}) = ∅ ↔ x ∈ ∼ b)
13	12	anbi1i 676	. . . . . 6 ⊢ (((b ∩ {x}) = ∅ ∧ A = (b ∪ {x})) ↔ (x ∈ ∼ b ∧ A = (b ∪ {x})))
14	8, 13	bitri 240	. . . . 5 ⊢ (∃y(y = {x} ∧ ((b ∩ y) = ∅ ∧ A = (b ∪ y))) ↔ (x ∈ ∼ b ∧ A = (b ∪ {x})))
15	14	exbii 1582	. . . 4 ⊢ (∃x∃y(y = {x} ∧ ((b ∩ y) = ∅ ∧ A = (b ∪ y))) ↔ ∃x(x ∈ ∼ b ∧ A = (b ∪ {x})))
16		df-rex 2621	. . . . 5 ⊢ (∃y ∈ 1c ((b ∩ y) = ∅ ∧ A = (b ∪ y)) ↔ ∃y(y ∈ 1c ∧ ((b ∩ y) = ∅ ∧ A = (b ∪ y))))
17		el1c 4140	. . . . . . . . 9 ⊢ (y ∈ 1c ↔ ∃x y = {x})
18	17	anbi1i 676	. . . . . . . 8 ⊢ ((y ∈ 1c ∧ ((b ∩ y) = ∅ ∧ A = (b ∪ y))) ↔ (∃x y = {x} ∧ ((b ∩ y) = ∅ ∧ A = (b ∪ y))))
19		19.41v 1901	. . . . . . . 8 ⊢ (∃x(y = {x} ∧ ((b ∩ y) = ∅ ∧ A = (b ∪ y))) ↔ (∃x y = {x} ∧ ((b ∩ y) = ∅ ∧ A = (b ∪ y))))
20	18, 19	bitr4i 243	. . . . . . 7 ⊢ ((y ∈ 1c ∧ ((b ∩ y) = ∅ ∧ A = (b ∪ y))) ↔ ∃x(y = {x} ∧ ((b ∩ y) = ∅ ∧ A = (b ∪ y))))
21	20	exbii 1582	. . . . . 6 ⊢ (∃y(y ∈ 1c ∧ ((b ∩ y) = ∅ ∧ A = (b ∪ y))) ↔ ∃y∃x(y = {x} ∧ ((b ∩ y) = ∅ ∧ A = (b ∪ y))))
22		excom 1741	. . . . . 6 ⊢ (∃y∃x(y = {x} ∧ ((b ∩ y) = ∅ ∧ A = (b ∪ y))) ↔ ∃x∃y(y = {x} ∧ ((b ∩ y) = ∅ ∧ A = (b ∪ y))))
23	21, 22	bitri 240	. . . . 5 ⊢ (∃y(y ∈ 1c ∧ ((b ∩ y) = ∅ ∧ A = (b ∪ y))) ↔ ∃x∃y(y = {x} ∧ ((b ∩ y) = ∅ ∧ A = (b ∪ y))))
24	16, 23	bitri 240	. . . 4 ⊢ (∃y ∈ 1c ((b ∩ y) = ∅ ∧ A = (b ∪ y)) ↔ ∃x∃y(y = {x} ∧ ((b ∩ y) = ∅ ∧ A = (b ∪ y))))
25		df-rex 2621	. . . 4 ⊢ (∃x ∈ ∼ bA = (b ∪ {x}) ↔ ∃x(x ∈ ∼ b ∧ A = (b ∪ {x})))
26	15, 24, 25	3bitr4i 268	. . 3 ⊢ (∃y ∈ 1c ((b ∩ y) = ∅ ∧ A = (b ∪ y)) ↔ ∃x ∈ ∼ bA = (b ∪ {x}))
27	26	rexbii 2640	. 2 ⊢ (∃b ∈ M ∃y ∈ 1c ((b ∩ y) = ∅ ∧ A = (b ∪ y)) ↔ ∃b ∈ M ∃x ∈ ∼ bA = (b ∪ {x}))
28	1, 27	bitri 240	1 ⊢ (A ∈ (M +c 1c) ↔ ∃b ∈ M ∃x ∈ ∼ bA = (b ∪ {x}))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616   ∼ ccompl 3206   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  {csn 3738  1_cc1c 4135   +_c 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