Theorem eladdc 4399

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 2868	. 2 ⊢ (A ∈ (M +c N) → A ∈ V)
2		id 19	. . . . . 6 ⊢ (A = (b ∪ c) → A = (b ∪ c))
3		vex 2863	. . . . . . 7 ⊢ b ∈ V
4		vex 2863	. . . . . . 7 ⊢ c ∈ V
5	3, 4	unex 4107	. . . . . 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 2735	. . 3 ⊢ (∃c ∈ N ((b ∩ c) = ∅ ∧ A = (b ∪ c)) → A ∈ V)
9	8	rexlimivw 2735	. 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 2658	. . 3 ⊢ (a = A → (∃b ∈ M ∃c ∈ N ((b ∩ c) = ∅ ∧ a = (b ∪ c)) ↔ ∃b ∈ M ∃c ∈ N ((b ∩ c) = ∅ ∧ A = (b ∪ c))))
13		df-addc 4379	. . 3 ⊢ (M +c N) = {a ∣ ∃b ∈ M ∃c ∈ N ((b ∩ c) = ∅ ∧ a = (b ∪ c))}
14	12, 13	elab2g 2988	. 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 2616  Vcvv 2860   ∪ cun 3208   ∩ cin 3209  ∅c0 3551   +_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

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-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-addc 4379

This theorem is referenced by:  eladdci  4400  0nelsuc  4401  addcid1  4406  elsuc  4414  addcass  4416  addcnul1  4453  tfindi  4497  evenfinex  4504  oddfinex  4505  sfinltfin  4536  vfinspsslem1  4551  addcfnex  5825  ncdisjun  6137  ce0addcnnul  6180  addlec  6209  taddc  6230  letc  6232  addcdi  6251