![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nnsdomel | Structured version Visualization version GIF version |
Description: Strict dominance and elementhood are the same for finite ordinals. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
Ref | Expression |
---|---|
nnsdomel | β’ ((π΄ β Ο β§ π΅ β Ο) β (π΄ β π΅ β π΄ βΊ π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardnn 9986 | . . 3 β’ (π΄ β Ο β (cardβπ΄) = π΄) | |
2 | cardnn 9986 | . . 3 β’ (π΅ β Ο β (cardβπ΅) = π΅) | |
3 | eleq12 2815 | . . 3 β’ (((cardβπ΄) = π΄ β§ (cardβπ΅) = π΅) β ((cardβπ΄) β (cardβπ΅) β π΄ β π΅)) | |
4 | 1, 2, 3 | syl2an 594 | . 2 β’ ((π΄ β Ο β§ π΅ β Ο) β ((cardβπ΄) β (cardβπ΅) β π΄ β π΅)) |
5 | nnon 7874 | . . . 4 β’ (π΄ β Ο β π΄ β On) | |
6 | onenon 9972 | . . . 4 β’ (π΄ β On β π΄ β dom card) | |
7 | 5, 6 | syl 17 | . . 3 β’ (π΄ β Ο β π΄ β dom card) |
8 | nnon 7874 | . . . 4 β’ (π΅ β Ο β π΅ β On) | |
9 | onenon 9972 | . . . 4 β’ (π΅ β On β π΅ β dom card) | |
10 | 8, 9 | syl 17 | . . 3 β’ (π΅ β Ο β π΅ β dom card) |
11 | cardsdom2 10011 | . . 3 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) β (cardβπ΅) β π΄ βΊ π΅)) | |
12 | 7, 10, 11 | syl2an 594 | . 2 β’ ((π΄ β Ο β§ π΅ β Ο) β ((cardβπ΄) β (cardβπ΅) β π΄ βΊ π΅)) |
13 | 4, 12 | bitr3d 280 | 1 β’ ((π΄ β Ο β§ π΅ β Ο) β (π΄ β π΅ β π΄ βΊ π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 class class class wbr 5143 dom cdm 5672 Oncon0 6364 βcfv 6543 Οcom 7868 βΊ csdm 8961 cardccrd 9958 |
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 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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-om 7869 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-card 9962 |
This theorem is referenced by: fin23lem27 10351 |
Copyright terms: Public domain | W3C validator |