Theorem cdavalt 4902

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 2766	. . . . . 6 |- {(/)} e. V
2		xpexg 3255	. . . . . 6 |- ((A e. V /\ {(/)} e. V) -> (A X. {(/)}) e. V)
3	1, 2	mpan2 695	. . . . 5 |- (A e. V -> (A X. {(/)}) e. V)
4		snex 2746	. . . . . 6 |- {1o} e. V
5		xpexg 3255	. . . . . 6 |- ((B e. V /\ {1o} e. V) -> (B X. {1o}) e. V)
6	4, 5	mpan2 695	. . . . 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 2869	. . . 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 3196	. . . . 5 |- (x = A -> (x X. {(/)}) = (A X. {(/)}))
11	10	uneq1d 2180	. . . 4 |- (x = A -> ((x X. {(/)}) u. (y X. {1o})) = ((A X. {(/)}) u. (y X. {1o})))
12		xpeq1 3196	. . . . 5 |- (y = B -> (y X. {1o}) = (B X. {1o}))
13	12	uneq2d 2181	. . . 4 |- (y = B -> ((A X. {(/)}) u. (y X. {1o})) = ((A X. {(/)}) u. (B X. {1o})))
14		df-cda 4901	. . . . 5 |- +c = {<.<.x, y>., z>. | z = ((x X. {(/)}) u. (y X. {1o}))}
15		visset 1810	. . . . . . . 8 |- x e. V
16		visset 1810	. . . . . . . 8 |- y e. V
17	15, 16	pm3.2i 285	. . . . . . 7 |- (x e. V /\ y e. V)
18	17	biantrur 724	. . . . . 6 |- (z = ((x X. {(/)}) u. (y X. {1o})) <-> ((x e. V /\ y e. V) /\ z = ((x X. {(/)}) u. (y X. {1o}))))
19	18	oprabbii 3992	. . . . 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 1493	. . . 4 |- +c = {<.<.x, y>., z>. | ((x e. V /\ y e. V) /\ z = ((x X. {(/)}) u. (y X. {1o})))}
21	11, 13, 20	oprabval2g 4022	. . 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 916	. 2 |- ((A e. V /\ B e. V) -> (A +c B) = ((A X. {(/)}) u. (B X. {1o})))
23		elisset 1814	. 2 |- (A e. C -> A e. V)
24		elisset 1814	. 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 955   e. wcel 957  Vcvv 1808   u. cun 2042  (/)c0 2277  {csn 2406   X. cxp 3164  (class class class)co 3958  {copab2 3959  1oc1o 4121   +c ccda 4900

This theorem is referenced by:  cdaval 4903  cdafi 4919

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fv 3194  df-opr 3960  df-oprab 3961  df-cda 4901

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