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Mirrors > Home > MPE Home > Th. List > alephord2 | Structured version Visualization version GIF version |
Description: Ordering property of the aleph function. Theorem 8A(a) of [Enderton] p. 213 and its converse. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 9-Feb-2013.) |
Ref | Expression |
---|---|
alephord2 | β’ ((π΄ β On β§ π΅ β On) β (π΄ β π΅ β (β΅βπ΄) β (β΅βπ΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alephord 10098 | . 2 β’ ((π΄ β On β§ π΅ β On) β (π΄ β π΅ β (β΅βπ΄) βΊ (β΅βπ΅))) | |
2 | alephon 10092 | . . . 4 β’ (β΅βπ΄) β On | |
3 | alephon 10092 | . . . . 5 β’ (β΅βπ΅) β On | |
4 | onenon 9972 | . . . . 5 β’ ((β΅βπ΅) β On β (β΅βπ΅) β dom card) | |
5 | 3, 4 | ax-mp 5 | . . . 4 β’ (β΅βπ΅) β dom card |
6 | cardsdomel 9997 | . . . 4 β’ (((β΅βπ΄) β On β§ (β΅βπ΅) β dom card) β ((β΅βπ΄) βΊ (β΅βπ΅) β (β΅βπ΄) β (cardβ(β΅βπ΅)))) | |
7 | 2, 5, 6 | mp2an 690 | . . 3 β’ ((β΅βπ΄) βΊ (β΅βπ΅) β (β΅βπ΄) β (cardβ(β΅βπ΅))) |
8 | alephcard 10093 | . . . 4 β’ (cardβ(β΅βπ΅)) = (β΅βπ΅) | |
9 | 8 | eleq2i 2817 | . . 3 β’ ((β΅βπ΄) β (cardβ(β΅βπ΅)) β (β΅βπ΄) β (β΅βπ΅)) |
10 | 7, 9 | bitri 274 | . 2 β’ ((β΅βπ΄) βΊ (β΅βπ΅) β (β΅βπ΄) β (β΅βπ΅)) |
11 | 1, 10 | bitrdi 286 | 1 β’ ((π΄ β On β§ π΅ β On) β (π΄ β π΅ β (β΅βπ΄) β (β΅βπ΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β wcel 2098 class class class wbr 5143 dom cdm 5672 Oncon0 6364 βcfv 6543 βΊ csdm 8961 cardccrd 9958 β΅cale 9959 |
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-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 |
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-rmo 3364 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-iun 4993 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-se 5628 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-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-isom 6552 df-riota 7372 df-ov 7419 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-oi 9533 df-har 9580 df-card 9962 df-aleph 9963 |
This theorem is referenced by: alephord2i 10100 alephord3 10101 alephiso 10121 alephval3 10133 alephiso2 43053 |
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