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Theorem ficardadju 10143
Description: The disjoint union of finite sets is equinumerous to the ordinal sum of the cardinalities of those sets. (Contributed by BTernaryTau, 3-Jul-2024.)
Assertion
Ref Expression
ficardadju ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))

Proof of Theorem ficardadju
StepHypRef Expression
1 ficardom 9905 . . . 4 (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
2 ficardom 9905 . . . 4 (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
3 nnadju 10141 . . . . 5 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → (card‘((card‘𝐴) ⊔ (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
4 df-dju 9845 . . . . . . 7 ((card‘𝐴) ⊔ (card‘𝐵)) = (({∅} × (card‘𝐴)) ∪ ({1o} × (card‘𝐵)))
5 snfi 8994 . . . . . . . . 9 {∅} ∈ Fin
6 nnfi 9117 . . . . . . . . 9 ((card‘𝐴) ∈ ω → (card‘𝐴) ∈ Fin)
7 xpfi 9267 . . . . . . . . 9 (({∅} ∈ Fin ∧ (card‘𝐴) ∈ Fin) → ({∅} × (card‘𝐴)) ∈ Fin)
85, 6, 7sylancr 588 . . . . . . . 8 ((card‘𝐴) ∈ ω → ({∅} × (card‘𝐴)) ∈ Fin)
9 snfi 8994 . . . . . . . . 9 {1o} ∈ Fin
10 nnfi 9117 . . . . . . . . 9 ((card‘𝐵) ∈ ω → (card‘𝐵) ∈ Fin)
11 xpfi 9267 . . . . . . . . 9 (({1o} ∈ Fin ∧ (card‘𝐵) ∈ Fin) → ({1o} × (card‘𝐵)) ∈ Fin)
129, 10, 11sylancr 588 . . . . . . . 8 ((card‘𝐵) ∈ ω → ({1o} × (card‘𝐵)) ∈ Fin)
13 unfi 9122 . . . . . . . 8 ((({∅} × (card‘𝐴)) ∈ Fin ∧ ({1o} × (card‘𝐵)) ∈ Fin) → (({∅} × (card‘𝐴)) ∪ ({1o} × (card‘𝐵))) ∈ Fin)
148, 12, 13syl2an 597 . . . . . . 7 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → (({∅} × (card‘𝐴)) ∪ ({1o} × (card‘𝐵))) ∈ Fin)
154, 14eqeltrid 2838 . . . . . 6 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) ⊔ (card‘𝐵)) ∈ Fin)
16 ficardid 9906 . . . . . 6 (((card‘𝐴) ⊔ (card‘𝐵)) ∈ Fin → (card‘((card‘𝐴) ⊔ (card‘𝐵))) ≈ ((card‘𝐴) ⊔ (card‘𝐵)))
1715, 16syl 17 . . . . 5 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → (card‘((card‘𝐴) ⊔ (card‘𝐵))) ≈ ((card‘𝐴) ⊔ (card‘𝐵)))
183, 17eqbrtrrd 5133 . . . 4 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +o (card‘𝐵)) ≈ ((card‘𝐴) ⊔ (card‘𝐵)))
191, 2, 18syl2an 597 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ≈ ((card‘𝐴) ⊔ (card‘𝐵)))
20 ficardid 9906 . . . 4 (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴)
21 ficardid 9906 . . . 4 (𝐵 ∈ Fin → (card‘𝐵) ≈ 𝐵)
22 djuen 10113 . . . 4 (((card‘𝐴) ≈ 𝐴 ∧ (card‘𝐵) ≈ 𝐵) → ((card‘𝐴) ⊔ (card‘𝐵)) ≈ (𝐴𝐵))
2320, 21, 22syl2an 597 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) ⊔ (card‘𝐵)) ≈ (𝐴𝐵))
24 entr 8952 . . 3 ((((card‘𝐴) +o (card‘𝐵)) ≈ ((card‘𝐴) ⊔ (card‘𝐵)) ∧ ((card‘𝐴) ⊔ (card‘𝐵)) ≈ (𝐴𝐵)) → ((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴𝐵))
2519, 23, 24syl2anc 585 . 2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴𝐵))
2625ensymd 8951 1 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  cun 3912  c0 4286  {csn 4590   class class class wbr 5109   × cxp 5635  cfv 6500  (class class class)co 7361  ωcom 7806  1oc1o 8409   +o coa 8413  cen 8886  Fincfn 8889  cdju 9842  cardccrd 9879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-oadd 8420  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-dju 9845  df-card 9883
This theorem is referenced by:  ficardun  10144  ficardun2  10146
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