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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 9058 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card) | |
2 | finnum 9058 | . . . . . . 7 ⊢ (𝐵 ∈ Fin → 𝐵 ∈ dom card) | |
3 | cardacda 9306 | . . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵))) | |
4 | 1, 2, 3 | syl2an 590 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵))) |
5 | 4 | 3adant3 1163 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵))) |
6 | 5 | ensymd 8244 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → ((card‘𝐴) +𝑜 (card‘𝐵)) ≈ (𝐴 +𝑐 𝐵)) |
7 | cdaun 9280 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 +𝑐 𝐵) ≈ (𝐴 ∪ 𝐵)) | |
8 | entr 8245 | . . . 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 580 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → ((card‘𝐴) +𝑜 (card‘𝐵)) ≈ (𝐴 ∪ 𝐵)) |
10 | carden2b 9077 | . . 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 9071 | . . . 4 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω) | |
13 | ficardom 9071 | . . . 4 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω) | |
14 | nnacl 7929 | . . . . 5 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω) | |
15 | cardnn 9073 | . . . . 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 590 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = ((card‘𝐴) +𝑜 (card‘𝐵))) |
18 | 17 | 3adant3 1163 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = ((card‘𝐴) +𝑜 (card‘𝐵))) |
19 | 11, 18 | eqtr3d 2833 | 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 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ∪ cun 3765 ∩ cin 3766 ∅c0 4113 class class class wbr 4841 dom cdm 5310 ‘cfv 6099 (class class class)co 6876 ωcom 7297 +𝑜 coa 7794 ≈ cen 8190 Fincfn 8193 cardccrd 9045 +𝑐 ccda 9275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-oadd 7801 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-card 9049 df-cda 9276 |
This theorem is referenced by: hashun 13417 |
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