Theorem cdavalt 4919

Description: Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258.

Assertion

Ref	Expression
cdavalt	|- ((A e. C /\ B e. D) -> (A +c B) = ((A X. {(/)}) u. (B X. {1o})))

Proof of Theorem cdavalt

Step	Hyp	Ref	Expression
1		p0ex 2770	. . . . . 6 |- {(/)} e. V
2		xpexg 3259	. . . . . 6 |- ((A e. V /\ {(/)} e. V) -> (A X. {(/)}) e. V)
3	1, 2	mpan2 696	. . . . 5 |- (A e. V -> (A X. {(/)}) e. V)
4		snex 2750	. . . . . 6 |- {1o} e. V
5		xpexg 3259	. . . . . 6 |- ((B e. V /\ {1o} e. V) -> (B X. {1o}) e. V)
6	4, 5	mpan2 696	. . . . 5 |- (B e. V -> (B X. {1o}) e. V)
7	3, 6	anim12i 333	. . . 4 |- ((A e. V /\ B e. V) -> ((A X. {(/)}) e. V /\ (B X. {1o}) e. V))
8		unexb 2873	. . . 4 |- (((A X. {(/)}) e. V /\ (B X. {1o}) e. V) <-> ((A X. {(/)}) u. (B X. {1o})) e. V)
9	7, 8	sylib 198	. . 3 |- ((A e. V /\ B e. V) -> ((A X. {(/)}) u. (B X. {1o})) e. V)
10		xpeq1 3200	. . . . 5 |- (x = A -> (x X. {(/)}) = (A X. {(/)}))
11	10	uneq1d 2183	. . . 4 |- (x = A -> ((x X. {(/)}) u. (y X. {1o})) = ((A X. {(/)}) u. (y X. {1o})))
12		xpeq1 3200	. . . . 5 |- (y = B -> (y X. {1o}) = (B X. {1o}))
13	12	uneq2d 2184	. . . 4 |- (y = B -> ((A X. {(/)}) u. (y X. {1o})) = ((A X. {(/)}) u. (B X. {1o})))
14		df-cda 4918	. . . . 5 |- +c = {<.<.x, y>., z>. | z = ((x X. {(/)}) u. (y X. {1o}))}
15		visset 1813	. . . . . . . 8 |- x e. V
16		visset 1813	. . . . . . . 8 |- y e. V
17	15, 16	pm3.2i 285	. . . . . . 7 |- (x e. V /\ y e. V)
18	17	biantrur 725	. . . . . 6 |- (z = ((x X. {(/)}) u. (y X. {1o})) <-> ((x e. V /\ y e. V) /\ z = ((x X. {(/)}) u. (y X. {1o}))))
19	18	oprabbii 3997	. . . . 5 |- {<.<.x, y>., z>. | z = ((x X. {(/)}) u. (y X. {1o}))} = {<.<.x, y>., z>. | ((x e. V /\ y e. V) /\ z = ((x X. {(/)}) u. (y X. {1o})))}
20	14, 19	eqtr 1495	. . . 4 |- +c = {<.<.x, y>., z>. | ((x e. V /\ y e. V) /\ z = ((x X. {(/)}) u. (y X. {1o})))}
21	11, 13, 20	oprabval2g 4027	. . 3 |- ((A e. V /\ B e. V /\ ((A X. {(/)}) u. (B X. {1o})) e. V) -> (A +c B) = ((A X. {(/)}) u. (B X. {1o})))
22	9, 21	mpd3an3 917	. 2 |- ((A e. V /\ B e. V) -> (A +c B) = ((A X. {(/)}) u. (B X. {1o})))
23		elisset 1817	. 2 |- (A e. C -> A e. V)
24		elisset 1817	. 2 |- (B e. D -> B e. V)
25	22, 23, 24	syl2an 454	1 |- ((A e. C /\ B e. D) -> (A +c B) = ((A X. {(/)}) u. (B X. {1o})))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811   u. cun 2045  (/)c0 2280  {csn 2409   X. cxp 3168  (class class class)co 3963  {copab2 3964  1oc1o 4128   +c ccda 4917

This theorem is referenced by:  cdaval 4920  cdafi 4936

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-opr 3965  df-oprab 3966  df-cda 4918

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