Theorem eladdc 4398

Description: Membership in cardinal addition. Theorem X.1.1 of [Rosser] p. 275. (Contributed by SF, 16-Jan-2015.)

Assertion

Ref	Expression
eladdc	⊢ (A ∈ (M +c N) ↔ ∃b ∈ M ∃c ∈ N ((b ∩ c) = ∅ ∧ A = (b ∪ c)))

Distinct variable groups:   A,b,c   M,b,c   N,b,c

Proof of Theorem eladdc

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2867	. 2 ⊢ (A ∈ (M +c N) → A ∈ V)
2		id 19	. . . . . 6 ⊢ (A = (b ∪ c) → A = (b ∪ c))
3		vex 2862	. . . . . . 7 ⊢ b ∈ V
4		vex 2862	. . . . . . 7 ⊢ c ∈ V
5	3, 4	unex 4106	. . . . . 6 ⊢ (b ∪ c) ∈ V
6	2, 5	syl6eqel 2441	. . . . 5 ⊢ (A = (b ∪ c) → A ∈ V)
7	6	adantl 452	. . . 4 ⊢ (((b ∩ c) = ∅ ∧ A = (b ∪ c)) → A ∈ V)
8	7	rexlimivw 2734	. . 3 ⊢ (∃c ∈ N ((b ∩ c) = ∅ ∧ A = (b ∪ c)) → A ∈ V)
9	8	rexlimivw 2734	. 2 ⊢ (∃b ∈ M ∃c ∈ N ((b ∩ c) = ∅ ∧ A = (b ∪ c)) → A ∈ V)
10		eqeq1 2359	. . . . 5 ⊢ (a = A → (a = (b ∪ c) ↔ A = (b ∪ c)))
11	10	anbi2d 684	. . . 4 ⊢ (a = A → (((b ∩ c) = ∅ ∧ a = (b ∪ c)) ↔ ((b ∩ c) = ∅ ∧ A = (b ∪ c))))
12	11	2rexbidv 2657	. . 3 ⊢ (a = A → (∃b ∈ M ∃c ∈ N ((b ∩ c) = ∅ ∧ a = (b ∪ c)) ↔ ∃b ∈ M ∃c ∈ N ((b ∩ c) = ∅ ∧ A = (b ∪ c))))
13		df-addc 4378	. . 3 ⊢ (M +c N) = {a ∣ ∃b ∈ M ∃c ∈ N ((b ∩ c) = ∅ ∧ a = (b ∪ c))}
14	12, 13	elab2g 2987	. 2 ⊢ (A ∈ V → (A ∈ (M +c N) ↔ ∃b ∈ M ∃c ∈ N ((b ∩ c) = ∅ ∧ A = (b ∪ c))))
15	1, 9, 14	pm5.21nii 342	1 ⊢ (A ∈ (M +c N) ↔ ∃b ∈ M ∃c ∈ N ((b ∩ c) = ∅ ∧ A = (b ∪ c)))

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859   ∪ cun 3207   ∩ cin 3208  ∅c0 3550   +_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

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-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-addc 4378

This theorem is referenced by:  eladdci  4399  0nelsuc  4400  addcid1  4405  elsuc  4413  addcass  4415  addcnul1  4452  tfindi  4496  evenfinex  4503  oddfinex  4504  sfinltfin  4535  vfinspsslem1  4550  addcfnex  5824  ncdisjun  6136  ce0addcnnul  6179  addlec  6208  taddc  6228  letc  6230  addcdi  6249