Theorem cdaval 4920

Description: Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258. For cardinal arithmetic, we follow Mendelson. Rather than defining operations restricted to cardinal numbers, we use this disjoint union operation for addition, while cross product and set exponentiation stand in for cardinal multiplication and exponentiation. Equinumerosity and dominance serve the roles of equality and ordering. If we wanted to, we could easily convert our theorems to actual cardinal number operations via carden 4831, carddom 4836, and cardsdom 4837. The advantage of Mendelson's approach is that we can directly use many equinumerosity theorems that we already have available.

Hypotheses

Ref	Expression
cdaval.1	|- A e. V
cdaval.2	|- B e. V

Assertion

Ref	Expression
cdaval	|- (A +c B) = ((A X. {(/)}) u. (B X. {1o}))

Proof of Theorem cdaval

Step	Hyp	Ref	Expression
1		cdaval.1	. 2 |- A e. V
2		cdaval.2	. 2 |- B e. V
3		cdavalt 4919	. 2 |- ((A e. V /\ B e. V) -> (A +c B) = ((A X. {(/)}) u. (B X. {1o})))
4	1, 2, 3	mp2an 697	1 |- (A +c B) = ((A X. {(/)}) u. (B X. {1o}))

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

This theorem is referenced by:  uncdadom 4921  cdaun 4922  pm110.643 4923  cdaen 4924  cda0en 4925  cda1en 4926  xp2cda 4928  cdacomen 4929  cdaassen 4930  xpcdaen 4931  mapcdaen 4932  cdadom1 4933  alephadd 7582

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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