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Mirrors > Home > MPE Home > Th. List > nnadjuALT | Structured version Visualization version GIF version |
Description: Shorter proof of nnadju 10141 using ax-rep 5246. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 6-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nnadjuALT | β’ ((π΄ β Ο β§ π΅ β Ο) β (cardβ(π΄ β π΅)) = (π΄ +o π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 7812 | . . . 4 β’ (π΄ β Ο β π΄ β On) | |
2 | nnon 7812 | . . . 4 β’ (π΅ β Ο β π΅ β On) | |
3 | onadju 10137 | . . . 4 β’ ((π΄ β On β§ π΅ β On) β (π΄ +o π΅) β (π΄ β π΅)) | |
4 | 1, 2, 3 | syl2an 597 | . . 3 β’ ((π΄ β Ο β§ π΅ β Ο) β (π΄ +o π΅) β (π΄ β π΅)) |
5 | carden2b 9911 | . . 3 β’ ((π΄ +o π΅) β (π΄ β π΅) β (cardβ(π΄ +o π΅)) = (cardβ(π΄ β π΅))) | |
6 | 4, 5 | syl 17 | . 2 β’ ((π΄ β Ο β§ π΅ β Ο) β (cardβ(π΄ +o π΅)) = (cardβ(π΄ β π΅))) |
7 | nnacl 8562 | . . 3 β’ ((π΄ β Ο β§ π΅ β Ο) β (π΄ +o π΅) β Ο) | |
8 | cardnn 9907 | . . 3 β’ ((π΄ +o π΅) β Ο β (cardβ(π΄ +o π΅)) = (π΄ +o π΅)) | |
9 | 7, 8 | syl 17 | . 2 β’ ((π΄ β Ο β§ π΅ β Ο) β (cardβ(π΄ +o π΅)) = (π΄ +o π΅)) |
10 | 6, 9 | eqtr3d 2775 | 1 β’ ((π΄ β Ο β§ π΅ β Ο) β (cardβ(π΄ β π΅)) = (π΄ +o π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 class class class wbr 5109 Oncon0 6321 βcfv 6500 (class class class)co 7361 Οcom 7806 +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-dom 8891 df-sdom 8892 df-fin 8893 df-dju 9845 df-card 9883 |
This theorem is referenced by: (None) |
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