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Theorem cardacda 9221
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 8969 . . . 4 (card‘𝐴) ∈ On
2 cardon 8969 . . . 4 (card‘𝐵) ∈ On
3 onacda 9220 . . . 4 (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) +𝑜 (card‘𝐵)) ≈ ((card‘𝐴) +𝑐 (card‘𝐵)))
41, 2, 3mp2an 664 . . 3 ((card‘𝐴) +𝑜 (card‘𝐵)) ≈ ((card‘𝐴) +𝑐 (card‘𝐵))
5 cardid2 8978 . . . 4 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
6 cardid2 8978 . . . 4 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
7 cdaen 9196 . . . 4 (((card‘𝐴) ≈ 𝐴 ∧ (card‘𝐵) ≈ 𝐵) → ((card‘𝐴) +𝑐 (card‘𝐵)) ≈ (𝐴 +𝑐 𝐵))
85, 6, 7syl2an 575 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) +𝑐 (card‘𝐵)) ≈ (𝐴 +𝑐 𝐵))
9 entr 8160 . . 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 567 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) +𝑜 (card‘𝐵)) ≈ (𝐴 +𝑐 𝐵))
1110ensymd 8159 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 382  wcel 2144   class class class wbr 4784  dom cdm 5249  Oncon0 5866  cfv 6031  (class class class)co 6792   +𝑜 coa 7709  cen 8105  cardccrd 8960   +𝑐 ccda 9190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-en 8109  df-card 8964  df-cda 9191
This theorem is referenced by:  cdanum  9222  ficardun  9225  ficardun2  9226  pwsdompw  9227
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