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Theorem cardacda 8972
Description: The cardinal sum is equinumerous to an ordinal sum of the cardinals. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
cardacda ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵)))

Proof of Theorem cardacda
StepHypRef Expression
1 cardon 8722 . . . 4 (card‘𝐴) ∈ On
2 cardon 8722 . . . 4 (card‘𝐵) ∈ On
3 onacda 8971 . . . 4 (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) +𝑜 (card‘𝐵)) ≈ ((card‘𝐴) +𝑐 (card‘𝐵)))
41, 2, 3mp2an 707 . . 3 ((card‘𝐴) +𝑜 (card‘𝐵)) ≈ ((card‘𝐴) +𝑐 (card‘𝐵))
5 cardid2 8731 . . . 4 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
6 cardid2 8731 . . . 4 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
7 cdaen 8947 . . . 4 (((card‘𝐴) ≈ 𝐴 ∧ (card‘𝐵) ≈ 𝐵) → ((card‘𝐴) +𝑐 (card‘𝐵)) ≈ (𝐴 +𝑐 𝐵))
85, 6, 7syl2an 494 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) +𝑐 (card‘𝐵)) ≈ (𝐴 +𝑐 𝐵))
9 entr 7960 . . 3 ((((card‘𝐴) +𝑜 (card‘𝐵)) ≈ ((card‘𝐴) +𝑐 (card‘𝐵)) ∧ ((card‘𝐴) +𝑐 (card‘𝐵)) ≈ (𝐴 +𝑐 𝐵)) → ((card‘𝐴) +𝑜 (card‘𝐵)) ≈ (𝐴 +𝑐 𝐵))
104, 8, 9sylancr 694 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) +𝑜 (card‘𝐵)) ≈ (𝐴 +𝑐 𝐵))
1110ensymd 7959 1 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987   class class class wbr 4618  dom cdm 5079  Oncon0 5687  cfv 5852  (class class class)co 6610   +𝑜 coa 7509  cen 7904  cardccrd 8713   +𝑐 ccda 8941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7908  df-card 8717  df-cda 8942
This theorem is referenced by:  cdanum  8973  ficardun  8976  ficardun2  8977  pwsdompw  8978
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