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Theorem cdaval 8571
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 Cartesian 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 8947, carddom 8950, and cardsdom 8951. The advantage of Mendelson's approach is that we can directly use many equinumerosity theorems that we already have available. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
Assertion
Ref Expression
cdaval

Proof of Theorem cdaval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3118 . 2
2 elex 3118 . 2
3 p0ex 4639 . . . . . 6
4 xpexg 6602 . . . . . 6
53, 4mpan2 671 . . . . 5
6 snex 4693 . . . . . 6
7 xpexg 6602 . . . . . 6
86, 7mpan2 671 . . . . 5
95, 8anim12i 566 . . . 4
10 unexb 6600 . . . 4
119, 10sylib 196 . . 3
12 xpeq1 5018 . . . . 5
1312uneq1d 3656 . . . 4
14 xpeq1 5018 . . . . 5
1514uneq2d 3657 . . . 4
16 df-cda 8569 . . . 4
1713, 15, 16ovmpt2g 6437 . . 3
1811, 17mpd3an3 1325 . 2
191, 2, 18syl2an 477 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818   cvv 3109  u.cun 3473   c0 3784  {csn 4029  X.cxp 5002  (class class class)co 6296   c1o 7142   ccda 8568
This theorem is referenced by:  uncdadom  8572  cdaun  8573  cdaen  8574  cda1dif  8577  pm110.643  8578  xp2cda  8581  cdacomen  8582  cdaassen  8583  xpcdaen  8584  mapcdaen  8585  cdadom1  8587  cdaxpdom  8590  cdafi  8591  cdainf  8593  infcda1  8594  pwcdadom  8617  isfin4-3  8716  alephadd  8973  canthp1lem2  9052  xpsc  14954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-cda 8569
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