![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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.) Avoid ax-rep 5276. (Revised by BTernaryTau, 3-Jul-2024.) |
Ref | Expression |
---|---|
ficardun | β’ ((π΄ β Fin β§ π΅ β Fin β§ (π΄ β© π΅) = β ) β (cardβ(π΄ βͺ π΅)) = ((cardβπ΄) +o (cardβπ΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ficardadju 10191 | . . . . . 6 β’ ((π΄ β Fin β§ π΅ β Fin) β (π΄ β π΅) β ((cardβπ΄) +o (cardβπ΅))) | |
2 | 1 | 3adant3 1129 | . . . . 5 β’ ((π΄ β Fin β§ π΅ β Fin β§ (π΄ β© π΅) = β ) β (π΄ β π΅) β ((cardβπ΄) +o (cardβπ΅))) |
3 | 2 | ensymd 8998 | . . . 4 β’ ((π΄ β Fin β§ π΅ β Fin β§ (π΄ β© π΅) = β ) β ((cardβπ΄) +o (cardβπ΅)) β (π΄ β π΅)) |
4 | endjudisj 10160 | . . . 4 β’ ((π΄ β Fin β§ π΅ β Fin β§ (π΄ β© π΅) = β ) β (π΄ β π΅) β (π΄ βͺ π΅)) | |
5 | entr 8999 | . . . 4 β’ ((((cardβπ΄) +o (cardβπ΅)) β (π΄ β π΅) β§ (π΄ β π΅) β (π΄ βͺ π΅)) β ((cardβπ΄) +o (cardβπ΅)) β (π΄ βͺ π΅)) | |
6 | 3, 4, 5 | syl2anc 583 | . . 3 β’ ((π΄ β Fin β§ π΅ β Fin β§ (π΄ β© π΅) = β ) β ((cardβπ΄) +o (cardβπ΅)) β (π΄ βͺ π΅)) |
7 | carden2b 9959 | . . 3 β’ (((cardβπ΄) +o (cardβπ΅)) β (π΄ βͺ π΅) β (cardβ((cardβπ΄) +o (cardβπ΅))) = (cardβ(π΄ βͺ π΅))) | |
8 | 6, 7 | syl 17 | . 2 β’ ((π΄ β Fin β§ π΅ β Fin β§ (π΄ β© π΅) = β ) β (cardβ((cardβπ΄) +o (cardβπ΅))) = (cardβ(π΄ βͺ π΅))) |
9 | ficardom 9953 | . . . 4 β’ (π΄ β Fin β (cardβπ΄) β Ο) | |
10 | ficardom 9953 | . . . 4 β’ (π΅ β Fin β (cardβπ΅) β Ο) | |
11 | nnacl 8607 | . . . . 5 β’ (((cardβπ΄) β Ο β§ (cardβπ΅) β Ο) β ((cardβπ΄) +o (cardβπ΅)) β Ο) | |
12 | cardnn 9955 | . . . . 5 β’ (((cardβπ΄) +o (cardβπ΅)) β Ο β (cardβ((cardβπ΄) +o (cardβπ΅))) = ((cardβπ΄) +o (cardβπ΅))) | |
13 | 11, 12 | syl 17 | . . . 4 β’ (((cardβπ΄) β Ο β§ (cardβπ΅) β Ο) β (cardβ((cardβπ΄) +o (cardβπ΅))) = ((cardβπ΄) +o (cardβπ΅))) |
14 | 9, 10, 13 | syl2an 595 | . . 3 β’ ((π΄ β Fin β§ π΅ β Fin) β (cardβ((cardβπ΄) +o (cardβπ΅))) = ((cardβπ΄) +o (cardβπ΅))) |
15 | 14 | 3adant3 1129 | . 2 β’ ((π΄ β Fin β§ π΅ β Fin β§ (π΄ β© π΅) = β ) β (cardβ((cardβπ΄) +o (cardβπ΅))) = ((cardβπ΄) +o (cardβπ΅))) |
16 | 8, 15 | eqtr3d 2766 | 1 β’ ((π΄ β Fin β§ π΅ β Fin β§ (π΄ β© π΅) = β ) β (cardβ(π΄ βͺ π΅)) = ((cardβπ΄) +o (cardβπ΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βͺ cun 3939 β© cin 3940 β c0 4315 class class class wbr 5139 βcfv 6534 (class class class)co 7402 Οcom 7849 +o coa 8459 β cen 8933 Fincfn 8936 β cdju 9890 cardccrd 9927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-dju 9893 df-card 9931 |
This theorem is referenced by: hashun 14343 |
Copyright terms: Public domain | W3C validator |