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Mirrors > Home > MPE Home > Th. List > cardadju | Structured version Visualization version GIF version |
Description: The cardinal sum is equinumerous to an ordinal sum of the cardinals. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
cardadju | β’ ((π΄ β dom card β§ π΅ β dom card) β (π΄ β π΅) β ((cardβπ΄) +o (cardβπ΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardon 9938 | . . . 4 β’ (cardβπ΄) β On | |
2 | cardon 9938 | . . . 4 β’ (cardβπ΅) β On | |
3 | onadju 10187 | . . . 4 β’ (((cardβπ΄) β On β§ (cardβπ΅) β On) β ((cardβπ΄) +o (cardβπ΅)) β ((cardβπ΄) β (cardβπ΅))) | |
4 | 1, 2, 3 | mp2an 690 | . . 3 β’ ((cardβπ΄) +o (cardβπ΅)) β ((cardβπ΄) β (cardβπ΅)) |
5 | cardid2 9947 | . . . 4 β’ (π΄ β dom card β (cardβπ΄) β π΄) | |
6 | cardid2 9947 | . . . 4 β’ (π΅ β dom card β (cardβπ΅) β π΅) | |
7 | djuen 10163 | . . . 4 β’ (((cardβπ΄) β π΄ β§ (cardβπ΅) β π΅) β ((cardβπ΄) β (cardβπ΅)) β (π΄ β π΅)) | |
8 | 5, 6, 7 | syl2an 596 | . . 3 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) β (cardβπ΅)) β (π΄ β π΅)) |
9 | entr 9001 | . . 3 β’ ((((cardβπ΄) +o (cardβπ΅)) β ((cardβπ΄) β (cardβπ΅)) β§ ((cardβπ΄) β (cardβπ΅)) β (π΄ β π΅)) β ((cardβπ΄) +o (cardβπ΅)) β (π΄ β π΅)) | |
10 | 4, 8, 9 | sylancr 587 | . 2 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) +o (cardβπ΅)) β (π΄ β π΅)) |
11 | 10 | ensymd 9000 | 1 β’ ((π΄ β dom card β§ π΅ β dom card) β (π΄ β π΅) β ((cardβπ΄) +o (cardβπ΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β wcel 2106 class class class wbr 5148 dom cdm 5676 Oncon0 6364 βcfv 6543 (class class class)co 7408 +o coa 8462 β cen 8935 β cdju 9892 cardccrd 9929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-oadd 8469 df-er 8702 df-en 8939 df-dju 9895 df-card 9933 |
This theorem is referenced by: djunum 10189 ficardunOLD 10195 ficardun2OLD 10197 pwsdompw 10198 |
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