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Mirrors > Home > MPE Home > Th. List > nnadjuALT | Structured version Visualization version GIF version |
Description: Shorter proof of nnadju 10226 using ax-rep 5287. (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 7880 | . . . 4 β’ (π΄ β Ο β π΄ β On) | |
2 | nnon 7880 | . . . 4 β’ (π΅ β Ο β π΅ β On) | |
3 | onadju 10222 | . . . 4 β’ ((π΄ β On β§ π΅ β On) β (π΄ +o π΅) β (π΄ β π΅)) | |
4 | 1, 2, 3 | syl2an 594 | . . 3 β’ ((π΄ β Ο β§ π΅ β Ο) β (π΄ +o π΅) β (π΄ β π΅)) |
5 | carden2b 9996 | . . 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 8636 | . . 3 β’ ((π΄ β Ο β§ π΅ β Ο) β (π΄ +o π΅) β Ο) | |
8 | cardnn 9992 | . . 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 2769 | 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 394 = wceq 1533 β wcel 2098 class class class wbr 5150 Oncon0 6372 βcfv 6551 (class class class)co 7424 Οcom 7874 +o coa 8488 β cen 8965 β cdju 9927 cardccrd 9964 |
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 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 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 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-oadd 8495 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-dju 9930 df-card 9968 |
This theorem is referenced by: (None) |
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