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Mirrors > Home > MPE Home > Th. List > onssr1 | Structured version Visualization version GIF version |
Description: Initial segments of the ordinals are contained in initial segments of the cumulative hierarchy. (Contributed by FL, 20-Apr-2011.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
onssr1 | β’ (π΄ β dom π 1 β π΄ β (π 1βπ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1funlim 9783 | . . . . . . . . . 10 β’ (Fun π 1 β§ Lim dom π 1) | |
2 | 1 | simpri 485 | . . . . . . . . 9 β’ Lim dom π 1 |
3 | limord 6423 | . . . . . . . . 9 β’ (Lim dom π 1 β Ord dom π 1) | |
4 | ordtr1 6406 | . . . . . . . . 9 β’ (Ord dom π 1 β ((π₯ β π΄ β§ π΄ β dom π 1) β π₯ β dom π 1)) | |
5 | 2, 3, 4 | mp2b 10 | . . . . . . . 8 β’ ((π₯ β π΄ β§ π΄ β dom π 1) β π₯ β dom π 1) |
6 | 5 | ancoms 458 | . . . . . . 7 β’ ((π΄ β dom π 1 β§ π₯ β π΄) β π₯ β dom π 1) |
7 | rankonidlem 9845 | . . . . . . 7 β’ (π₯ β dom π 1 β (π₯ β βͺ (π 1 β On) β§ (rankβπ₯) = π₯)) | |
8 | 6, 7 | syl 17 | . . . . . 6 β’ ((π΄ β dom π 1 β§ π₯ β π΄) β (π₯ β βͺ (π 1 β On) β§ (rankβπ₯) = π₯)) |
9 | 8 | simprd 495 | . . . . 5 β’ ((π΄ β dom π 1 β§ π₯ β π΄) β (rankβπ₯) = π₯) |
10 | simpr 484 | . . . . 5 β’ ((π΄ β dom π 1 β§ π₯ β π΄) β π₯ β π΄) | |
11 | 9, 10 | eqeltrd 2829 | . . . 4 β’ ((π΄ β dom π 1 β§ π₯ β π΄) β (rankβπ₯) β π΄) |
12 | 8 | simpld 494 | . . . . 5 β’ ((π΄ β dom π 1 β§ π₯ β π΄) β π₯ β βͺ (π 1 β On)) |
13 | simpl 482 | . . . . 5 β’ ((π΄ β dom π 1 β§ π₯ β π΄) β π΄ β dom π 1) | |
14 | rankr1ag 9819 | . . . . 5 β’ ((π₯ β βͺ (π 1 β On) β§ π΄ β dom π 1) β (π₯ β (π 1βπ΄) β (rankβπ₯) β π΄)) | |
15 | 12, 13, 14 | syl2anc 583 | . . . 4 β’ ((π΄ β dom π 1 β§ π₯ β π΄) β (π₯ β (π 1βπ΄) β (rankβπ₯) β π΄)) |
16 | 11, 15 | mpbird 257 | . . 3 β’ ((π΄ β dom π 1 β§ π₯ β π΄) β π₯ β (π 1βπ΄)) |
17 | 16 | ex 412 | . 2 β’ (π΄ β dom π 1 β (π₯ β π΄ β π₯ β (π 1βπ΄))) |
18 | 17 | ssrdv 3984 | 1 β’ (π΄ β dom π 1 β π΄ β (π 1βπ΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 β wss 3945 βͺ cuni 4903 dom cdm 5672 β cima 5675 Ord word 6362 Oncon0 6363 Lim wlim 6364 Fun wfun 6536 βcfv 6542 π 1cr1 9779 rankcrnk 9780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 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 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-r1 9781 df-rank 9782 |
This theorem is referenced by: rankr1id 9879 ackbij2 10260 wunom 10737 r1limwun 10753 inar1 10792 r1tskina 10799 r1rankcld 43662 |
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