Theorem cdavalt 4931

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 2776	. . . . . 6 |- {(/)} e. V
2		xpexg 3265	. . . . . 6 |- ((A e. V /\ {(/)} e. V) -> (A X. {(/)}) e. V)
3	1, 2	mpan2 698	. . . . 5 |- (A e. V -> (A X. {(/)}) e. V)
4		snex 2756	. . . . . 6 |- {1o} e. V
5		xpexg 3265	. . . . . 6 |- ((B e. V /\ {1o} e. V) -> (B X. {1o}) e. V)
6	4, 5	mpan2 698	. . . . 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 2879	. . . 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 3206	. . . . 5 |- (x = A -> (x X. {(/)}) = (A X. {(/)}))
11	10	uneq1d 2186	. . . 4 |- (x = A -> ((x X. {(/)}) u. (y X. {1o})) = ((A X. {(/)}) u. (y X. {1o})))
12		xpeq1 3206	. . . . 5 |- (y = B -> (y X. {1o}) = (B X. {1o}))
13	12	uneq2d 2187	. . . 4 |- (y = B -> ((A X. {(/)}) u. (y X. {1o})) = ((A X. {(/)}) u. (B X. {1o})))
14		df-cda 4930	. . . . 5 |- +c = {<.<.x, y>., z>. | z = ((x X. {(/)}) u. (y X. {1o}))}
15		visset 1816	. . . . . . . 8 |- x e. V
16		visset 1816	. . . . . . . 8 |- y e. V
17	15, 16	pm3.2i 285	. . . . . . 7 |- (x e. V /\ y e. V)
18	17	biantrur 727	. . . . . 6 |- (z = ((x X. {(/)}) u. (y X. {1o})) <-> ((x e. V /\ y e. V) /\ z = ((x X. {(/)}) u. (y X. {1o}))))
19	18	oprabbii 4003	. . . . 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 1498	. . . 4 |- +c = {<.<.x, y>., z>. | ((x e. V /\ y e. V) /\ z = ((x X. {(/)}) u. (y X. {1o})))}
21	11, 13, 20	oprabval2g 4033	. . 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 919	. 2 |- ((A e. V /\ B e. V) -> (A +c B) = ((A X. {(/)}) u. (B X. {1o})))
23		elisset 1820	. 2 |- (A e. C -> A e. V)
24		elisset 1820	. 2 |- (B e. D -> B e. V)
25	22, 23, 24	syl2an 456	1 |- ((A e. C /\ B e. D) -> (A +c B) = ((A X. {(/)}) u. (B X. {1o})))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814   u. cun 2048  (/)c0 2283  {csn 2413   X. cxp 3174  (class class class)co 3969  {copab2 3970  1oc1o 4134   +c ccda 4929

This theorem is referenced by:  cdaval 4932  cdafi 4948

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-opr 3971  df-oprab 3972  df-cda 4930

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