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Mirrors > Home > MPE Home > Th. List > hsmexlem9 | Structured version Visualization version GIF version |
Description: Lemma for hsmex 10409. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
Ref | Expression |
---|---|
hsmexlem7.h | β’ π» = (rec((π§ β V β¦ (harβπ« (π Γ π§))), (harβπ« π)) βΎ Ο) |
Ref | Expression |
---|---|
hsmexlem9 | β’ (π β Ο β (π»βπ) β On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0suc 7868 | . 2 β’ (π β Ο β (π = β β¨ βπ β Ο π = suc π)) | |
2 | fveq2 6878 | . . . 4 β’ (π = β β (π»βπ) = (π»ββ )) | |
3 | hsmexlem7.h | . . . . . 6 β’ π» = (rec((π§ β V β¦ (harβπ« (π Γ π§))), (harβπ« π)) βΎ Ο) | |
4 | 3 | hsmexlem7 10400 | . . . . 5 β’ (π»ββ ) = (harβπ« π) |
5 | harcl 9536 | . . . . 5 β’ (harβπ« π) β On | |
6 | 4, 5 | eqeltri 2828 | . . . 4 β’ (π»ββ ) β On |
7 | 2, 6 | eqeltrdi 2840 | . . 3 β’ (π = β β (π»βπ) β On) |
8 | 3 | hsmexlem8 10401 | . . . . . 6 β’ (π β Ο β (π»βsuc π) = (harβπ« (π Γ (π»βπ)))) |
9 | harcl 9536 | . . . . . 6 β’ (harβπ« (π Γ (π»βπ))) β On | |
10 | 8, 9 | eqeltrdi 2840 | . . . . 5 β’ (π β Ο β (π»βsuc π) β On) |
11 | fveq2 6878 | . . . . . 6 β’ (π = suc π β (π»βπ) = (π»βsuc π)) | |
12 | 11 | eleq1d 2817 | . . . . 5 β’ (π = suc π β ((π»βπ) β On β (π»βsuc π) β On)) |
13 | 10, 12 | syl5ibrcom 246 | . . . 4 β’ (π β Ο β (π = suc π β (π»βπ) β On)) |
14 | 13 | rexlimiv 3147 | . . 3 β’ (βπ β Ο π = suc π β (π»βπ) β On) |
15 | 7, 14 | jaoi 855 | . 2 β’ ((π = β β¨ βπ β Ο π = suc π) β (π»βπ) β On) |
16 | 1, 15 | syl 17 | 1 β’ (π β Ο β (π»βπ) β On) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β¨ wo 845 = wceq 1541 β wcel 2106 βwrex 3069 Vcvv 3473 β c0 4318 π« cpw 4596 β¦ cmpt 5224 Γ cxp 5667 βΎ cres 5671 Oncon0 6353 suc csuc 6355 βcfv 6532 Οcom 7838 reccrdg 8391 harchar 9533 |
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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-om 7839 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-en 8923 df-dom 8924 df-oi 9487 df-har 9534 |
This theorem is referenced by: hsmexlem4 10406 hsmexlem5 10407 |
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