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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.) Avoid ax-rep 5285. (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 10222 | . . . . . 6 β’ ((π΄ β Fin β§ π΅ β Fin) β (π΄ β π΅) β ((cardβπ΄) +o (cardβπ΅))) | |
2 | 1 | 3adant3 1130 | . . . . 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 9025 | . . . 4 β’ ((π΄ β Fin β§ π΅ β Fin β§ (π΄ β© π΅) = β ) β ((cardβπ΄) +o (cardβπ΅)) β (π΄ β π΅)) |
4 | endjudisj 10191 | . . . 4 β’ ((π΄ β Fin β§ π΅ β Fin β§ (π΄ β© π΅) = β ) β (π΄ β π΅) β (π΄ βͺ π΅)) | |
5 | entr 9026 | . . . 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 9990 | . . 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 9984 | . . . 4 β’ (π΄ β Fin β (cardβπ΄) β Ο) | |
10 | ficardom 9984 | . . . 4 β’ (π΅ β Fin β (cardβπ΅) β Ο) | |
11 | nnacl 8631 | . . . . 5 β’ (((cardβπ΄) β Ο β§ (cardβπ΅) β Ο) β ((cardβπ΄) +o (cardβπ΅)) β Ο) | |
12 | cardnn 9986 | . . . . 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 1130 | . 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 2770 | 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 1085 = wceq 1534 β wcel 2099 βͺ cun 3945 β© cin 3946 β c0 4323 class class class wbr 5148 βcfv 6548 (class class class)co 7420 Οcom 7870 +o coa 8483 β cen 8960 Fincfn 8963 β cdju 9921 cardccrd 9958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-oadd 8490 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-dju 9924 df-card 9962 |
This theorem is referenced by: hashun 14373 |
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