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.

Step	Hyp	Ref	Expression
1		eladdc 4398	. 2 ⊢ (A ∈ (M +c 1c) ↔ ∃b ∈ M ∃y ∈ 1c ((b ∩ y) = ∅ ∧ A = (b ∪ y)))
2		snex 4111	. . . . . . 7 ⊢ {x} ∈ V
3		ineq2 3451	. . . . . . . . 9 ⊢ (y = {x} → (b ∩ y) = (b ∩ {x}))
4	3	eqeq1d 2361	. . . . . . . 8 ⊢ (y = {x} → ((b ∩ y) = ∅ ↔ (b ∩ {x}) = ∅))
5		uneq2 3412	. . . . . . . . 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 2894	. . . . . 6 ⊢ (∃y(y = {x} ∧ ((b ∩ y) = ∅ ∧ A = (b ∪ y))) ↔ ((b ∩ {x}) = ∅ ∧ A = (b ∪ {x})))
9		disjsn 3786	. . . . . . . 8 ⊢ ((b ∩ {x}) = ∅ ↔ ¬ x ∈ b)
10		vex 2862	. . . . . . . . 9 ⊢ x ∈ V
11	10	elcompl 3225	. . . . . . . 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 2620	. . . . 5 ⊢ (∃y ∈ 1c ((b ∩ y) = ∅ ∧ A = (b ∪ y)) ↔ ∃y(y ∈ 1c ∧ ((b ∩ y) = ∅ ∧ A = (b ∪ y))))
17		el1c 4139	. . . . . . . . 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 2620	. . . 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 2639	. 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 2615   ∼ ccompl 3205   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  {csn 3737  1_cc1c 4134   +_c cplc 4375

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 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