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Theorem cardadju 10171
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
cardadju ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))

Proof of Theorem cardadju
StepHypRef Expression
1 cardon 9921 . . . 4 (card‘𝐴) ∈ On
2 cardon 9921 . . . 4 (card‘𝐵) ∈ On
3 onadju 10170 . . . 4 (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) +o (card‘𝐵)) ≈ ((card‘𝐴) ⊔ (card‘𝐵)))
41, 2, 3mp2an 690 . . 3 ((card‘𝐴) +o (card‘𝐵)) ≈ ((card‘𝐴) ⊔ (card‘𝐵))
5 cardid2 9930 . . . 4 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
6 cardid2 9930 . . . 4 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
7 djuen 10146 . . . 4 (((card‘𝐴) ≈ 𝐴 ∧ (card‘𝐵) ≈ 𝐵) → ((card‘𝐴) ⊔ (card‘𝐵)) ≈ (𝐴𝐵))
85, 6, 7syl2an 596 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊔ (card‘𝐵)) ≈ (𝐴𝐵))
9 entr 8985 . . 3 ((((card‘𝐴) +o (card‘𝐵)) ≈ ((card‘𝐴) ⊔ (card‘𝐵)) ∧ ((card‘𝐴) ⊔ (card‘𝐵)) ≈ (𝐴𝐵)) → ((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴𝐵))
104, 8, 9sylancr 587 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴𝐵))
1110ensymd 8984 1 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106   class class class wbr 5141  dom cdm 5669  Oncon0 6353  cfv 6532  (class class class)co 7393   +o coa 8445  cen 8919  cdju 9875  cardccrd 9912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-oadd 8452  df-er 8686  df-en 8923  df-dju 9878  df-card 9916
This theorem is referenced by:  djunum  10172  ficardunOLD  10178  ficardun2OLD  10180  pwsdompw  10181
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