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Mirrors > Home > MPE Home > Th. List > ficardun | Structured version Visualization version GIF version |
Description: The cardinality of the union of disjoint, finite sets is the ordinal sum of their cardinalities. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
ficardun | ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘(𝐴 ∪ 𝐵)) = ((card‘𝐴) +𝑜 (card‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | finnum 8975 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card) | |
2 | finnum 8975 | . . . . . . 7 ⊢ (𝐵 ∈ Fin → 𝐵 ∈ dom card) | |
3 | cardacda 9223 | . . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵))) | |
4 | 1, 2, 3 | syl2an 577 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵))) |
5 | 4 | 3adant3 1126 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵))) |
6 | 5 | ensymd 8161 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → ((card‘𝐴) +𝑜 (card‘𝐵)) ≈ (𝐴 +𝑐 𝐵)) |
7 | cdaun 9197 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 +𝑐 𝐵) ≈ (𝐴 ∪ 𝐵)) | |
8 | entr 8162 | . . . 4 ⊢ ((((card‘𝐴) +𝑜 (card‘𝐵)) ≈ (𝐴 +𝑐 𝐵) ∧ (𝐴 +𝑐 𝐵) ≈ (𝐴 ∪ 𝐵)) → ((card‘𝐴) +𝑜 (card‘𝐵)) ≈ (𝐴 ∪ 𝐵)) | |
9 | 6, 7, 8 | syl2anc 567 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → ((card‘𝐴) +𝑜 (card‘𝐵)) ≈ (𝐴 ∪ 𝐵)) |
10 | carden2b 8994 | . . 3 ⊢ (((card‘𝐴) +𝑜 (card‘𝐵)) ≈ (𝐴 ∪ 𝐵) → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = (card‘(𝐴 ∪ 𝐵))) | |
11 | 9, 10 | syl 17 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = (card‘(𝐴 ∪ 𝐵))) |
12 | ficardom 8988 | . . . 4 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω) | |
13 | ficardom 8988 | . . . 4 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω) | |
14 | nnacl 7846 | . . . . 5 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω) | |
15 | cardnn 8990 | . . . . 5 ⊢ (((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = ((card‘𝐴) +𝑜 (card‘𝐵))) | |
16 | 14, 15 | syl 17 | . . . 4 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = ((card‘𝐴) +𝑜 (card‘𝐵))) |
17 | 12, 13, 16 | syl2an 577 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = ((card‘𝐴) +𝑜 (card‘𝐵))) |
18 | 17 | 3adant3 1126 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = ((card‘𝐴) +𝑜 (card‘𝐵))) |
19 | 11, 18 | eqtr3d 2807 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘(𝐴 ∪ 𝐵)) = ((card‘𝐴) +𝑜 (card‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ∪ cun 3722 ∩ cin 3723 ∅c0 4064 class class class wbr 4787 dom cdm 5250 ‘cfv 6032 (class class class)co 6794 ωcom 7213 +𝑜 coa 7711 ≈ cen 8107 Fincfn 8110 cardccrd 8962 +𝑐 ccda 9192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5824 df-ord 5870 df-on 5871 df-lim 5872 df-suc 5873 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-om 7214 df-wrecs 7560 df-recs 7622 df-rdg 7660 df-1o 7714 df-oadd 7718 df-er 7897 df-en 8111 df-dom 8112 df-sdom 8113 df-fin 8114 df-card 8966 df-cda 9193 |
This theorem is referenced by: hashun 13374 |
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