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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 9888 | . . . 4 β’ (cardβπ΄) β On | |
2 | cardon 9888 | . . . 4 β’ (cardβπ΅) β On | |
3 | onadju 10137 | . . . 4 β’ (((cardβπ΄) β On β§ (cardβπ΅) β On) β ((cardβπ΄) +o (cardβπ΅)) β ((cardβπ΄) β (cardβπ΅))) | |
4 | 1, 2, 3 | mp2an 691 | . . 3 β’ ((cardβπ΄) +o (cardβπ΅)) β ((cardβπ΄) β (cardβπ΅)) |
5 | cardid2 9897 | . . . 4 β’ (π΄ β dom card β (cardβπ΄) β π΄) | |
6 | cardid2 9897 | . . . 4 β’ (π΅ β dom card β (cardβπ΅) β π΅) | |
7 | djuen 10113 | . . . 4 β’ (((cardβπ΄) β π΄ β§ (cardβπ΅) β π΅) β ((cardβπ΄) β (cardβπ΅)) β (π΄ β π΅)) | |
8 | 5, 6, 7 | syl2an 597 | . . 3 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) β (cardβπ΅)) β (π΄ β π΅)) |
9 | entr 8952 | . . 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 588 | . 2 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) +o (cardβπ΅)) β (π΄ β π΅)) |
11 | 10 | ensymd 8951 | 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 397 β wcel 2107 class class class wbr 5109 dom cdm 5637 Oncon0 6321 βcfv 6500 (class class class)co 7361 +o coa 8413 β cen 8886 β cdju 9842 cardccrd 9879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-oadd 8420 df-er 8654 df-en 8890 df-dju 9845 df-card 9883 |
This theorem is referenced by: djunum 10139 ficardunOLD 10145 ficardun2OLD 10147 pwsdompw 10148 |
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