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Mirrors > Home > MPE Home > Th. List > newf | Structured version Visualization version GIF version |
Description: The new function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.) |
Ref | Expression |
---|---|
newf | β’ N :OnβΆπ« No |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-new 27341 | . 2 β’ N = (π₯ β On β¦ (( M βπ₯) β ( O βπ₯))) | |
2 | madef 27348 | . . . . . 6 β’ M :OnβΆπ« No | |
3 | 2 | ffvelcdmi 7085 | . . . . 5 β’ (π₯ β On β ( M βπ₯) β π« No ) |
4 | 3 | elpwid 4611 | . . . 4 β’ (π₯ β On β ( M βπ₯) β No ) |
5 | 4 | ssdifssd 4142 | . . 3 β’ (π₯ β On β (( M βπ₯) β ( O βπ₯)) β No ) |
6 | fvex 6904 | . . . . 5 β’ ( M βπ₯) β V | |
7 | 6 | difexi 5328 | . . . 4 β’ (( M βπ₯) β ( O βπ₯)) β V |
8 | 7 | elpw 4606 | . . 3 β’ ((( M βπ₯) β ( O βπ₯)) β π« No β (( M βπ₯) β ( O βπ₯)) β No ) |
9 | 5, 8 | sylibr 233 | . 2 β’ (π₯ β On β (( M βπ₯) β ( O βπ₯)) β π« No ) |
10 | 1, 9 | fmpti 7111 | 1 β’ N :OnβΆπ« No |
Colors of variables: wff setvar class |
Syntax hints: β wcel 2106 β cdif 3945 β wss 3948 π« cpw 4602 Oncon0 6364 βΆwf 6539 βcfv 6543 No csur 27140 M cmade 27334 O cold 27335 N cnew 27336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 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-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-1o 8465 df-2o 8466 df-no 27143 df-slt 27144 df-bday 27145 df-sslt 27280 df-scut 27282 df-made 27339 df-new 27341 |
This theorem is referenced by: newssno 27354 |
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