Theorem eladdci 4400

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

Assertion

Ref	Expression
eladdci	|- ((A e. M /\ B e. N /\ (A i^i B) = (/)) -> (A u. B) e. (M +o 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 u. B) = (A u. B)
2		ineq1 3451	. . . . . . . 8 |- (a = A -> (a i^i b) = (A i^i b))
3	2	eqeq1d 2361	. . . . . . 7 |- (a = A -> ((a i^i b) = (/) <-> (A i^i b) = (/)))
4		uneq1 3412	. . . . . . . 8 |- (a = A -> (a u. b) = (A u. b))
5	4	eqeq2d 2364	. . . . . . 7 |- (a = A -> ((A u. B) = (a u. b) <-> (A u. B) = (A u. b)))
6	3, 5	anbi12d 691	. . . . . 6 |- (a = A -> (((a i^i b) = (/) /\ (A u. B) = (a u. b)) <-> ((A i^i b) = (/) /\ (A u. B) = (A u. b))))
7		ineq2 3452	. . . . . . . 8 |- (b = B -> (A i^i b) = (A i^i B))
8	7	eqeq1d 2361	. . . . . . 7 |- (b = B -> ((A i^i b) = (/) <-> (A i^i B) = (/)))
9		uneq2 3413	. . . . . . . 8 |- (b = B -> (A u. b) = (A u. B))
10	9	eqeq2d 2364	. . . . . . 7 |- (b = B -> ((A u. B) = (A u. b) <-> (A u. B) = (A u. B)))
11	8, 10	anbi12d 691	. . . . . 6 |- (b = B -> (((A i^i b) = (/) /\ (A u. B) = (A u. b)) <-> ((A i^i B) = (/) /\ (A u. B) = (A u. B))))
12	6, 11	rspc2ev 2964	. . . . 5 |- ((A e. M /\ B e. N /\ ((A i^i B) = (/) /\ (A u. B) = (A u. B))) -> E.a e. M E.b e. N ((a i^i b) = (/) /\ (A u. B) = (a u. b)))
13	12	3expa 1151	. . . 4 |- (((A e. M /\ B e. N) /\ ((A i^i B) = (/) /\ (A u. B) = (A u. B))) -> E.a e. M E.b e. N ((a i^i b) = (/) /\ (A u. B) = (a u. b)))
14	1, 13	mpanr2 665	. . 3 |- (((A e. M /\ B e. N) /\ (A i^i B) = (/)) -> E.a e. M E.b e. N ((a i^i b) = (/) /\ (A u. B) = (a u. b)))
15	14	3impa 1146	. 2 |- ((A e. M /\ B e. N /\ (A i^i B) = (/)) -> E.a e. M E.b e. N ((a i^i b) = (/) /\ (A u. B) = (a u. b)))
16		eladdc 4399	. 2 |- ((A u. B) e. (M +o N) <-> E.a e. M E.b e. N ((a i^i b) = (/) /\ (A u. B) = (a u. b)))
17	15, 16	sylibr 203	1 |- ((A e. M /\ B e. N /\ (A i^i B) = (/)) -> (A u. B) e. (M +o N))

Syntax hints:   -> wi 4   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710  E.wrex 2616   u. cun 3208   i^i cin 3209  (/)c0 3551   +o 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-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 2479  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-addc 4379

This theorem is referenced by:  ncfindi  4476  tfindi  4497  nnadjoinpw  4522  sfinltfin  4536  ncdisjun  6137  tcdi  6165  ce0addcnnul  6180