Theorem eladdci 4399

Description: Inference form of membership in cardinal addition. (Contributed by SF, 26-Jan-2015.)

Assertion

Ref	Expression
eladdci	⊢ ((A ∈ M ∧ B ∈ N ∧ (A ∩ B) = ∅) → (A ∪ B) ∈ (M +c N))

Proof of Theorem eladdci

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2353	. . . 4 ⊢ (A ∪ B) = (A ∪ B)
2		ineq1 3450	. . . . . . . 8 ⊢ (a = A → (a ∩ b) = (A ∩ b))
3	2	eqeq1d 2361	. . . . . . 7 ⊢ (a = A → ((a ∩ b) = ∅ ↔ (A ∩ b) = ∅))
4		uneq1 3411	. . . . . . . 8 ⊢ (a = A → (a ∪ b) = (A ∪ b))
5	4	eqeq2d 2364	. . . . . . 7 ⊢ (a = A → ((A ∪ B) = (a ∪ b) ↔ (A ∪ B) = (A ∪ b)))
6	3, 5	anbi12d 691	. . . . . 6 ⊢ (a = A → (((a ∩ b) = ∅ ∧ (A ∪ B) = (a ∪ b)) ↔ ((A ∩ b) = ∅ ∧ (A ∪ B) = (A ∪ b))))
7		ineq2 3451	. . . . . . . 8 ⊢ (b = B → (A ∩ b) = (A ∩ B))
8	7	eqeq1d 2361	. . . . . . 7 ⊢ (b = B → ((A ∩ b) = ∅ ↔ (A ∩ B) = ∅))
9		uneq2 3412	. . . . . . . 8 ⊢ (b = B → (A ∪ b) = (A ∪ B))
10	9	eqeq2d 2364	. . . . . . 7 ⊢ (b = B → ((A ∪ B) = (A ∪ b) ↔ (A ∪ B) = (A ∪ B)))
11	8, 10	anbi12d 691	. . . . . 6 ⊢ (b = B → (((A ∩ b) = ∅ ∧ (A ∪ B) = (A ∪ b)) ↔ ((A ∩ B) = ∅ ∧ (A ∪ B) = (A ∪ B))))
12	6, 11	rspc2ev 2963	. . . . 5 ⊢ ((A ∈ M ∧ B ∈ N ∧ ((A ∩ B) = ∅ ∧ (A ∪ B) = (A ∪ B))) → ∃a ∈ M ∃b ∈ N ((a ∩ b) = ∅ ∧ (A ∪ B) = (a ∪ b)))
13	12	3expa 1151	. . . 4 ⊢ (((A ∈ M ∧ B ∈ N) ∧ ((A ∩ B) = ∅ ∧ (A ∪ B) = (A ∪ B))) → ∃a ∈ M ∃b ∈ N ((a ∩ b) = ∅ ∧ (A ∪ B) = (a ∪ b)))
14	1, 13	mpanr2 665	. . 3 ⊢ (((A ∈ M ∧ B ∈ N) ∧ (A ∩ B) = ∅) → ∃a ∈ M ∃b ∈ N ((a ∩ b) = ∅ ∧ (A ∪ B) = (a ∪ b)))
15	14	3impa 1146	. 2 ⊢ ((A ∈ M ∧ B ∈ N ∧ (A ∩ B) = ∅) → ∃a ∈ M ∃b ∈ N ((a ∩ b) = ∅ ∧ (A ∪ B) = (a ∪ b)))
16		eladdc 4398	. 2 ⊢ ((A ∪ B) ∈ (M +c N) ↔ ∃a ∈ M ∃b ∈ N ((a ∩ b) = ∅ ∧ (A ∪ B) = (a ∪ b)))
17	15, 16	sylibr 203	1 ⊢ ((A ∈ M ∧ B ∈ N ∧ (A ∩ B) = ∅) → (A ∪ B) ∈ (M +c N))

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∪ cun 3207   ∩ cin 3208  ∅c0 3550   +_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

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-in 3213  df-un 3214  df-addc 4378

This theorem is referenced by:  ncfindi  4475  tfindi  4496  nnadjoinpw  4521  sfinltfin  4535  ncdisjun  6136  tcdi  6164  ce0addcnnul  6179