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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 9758 | . . . . . . . . . 10 β’ (Fun π 1 β§ Lim dom π 1) | |
2 | 1 | simpri 485 | . . . . . . . . 9 β’ Lim dom π 1 |
3 | limord 6415 | . . . . . . . . 9 β’ (Lim dom π 1 β Ord dom π 1) | |
4 | ordtr1 6398 | . . . . . . . . 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 9820 | . . . . . . 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 2825 | . . . 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 9794 | . . . . 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 3981 | 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 1533 β wcel 2098 β wss 3941 βͺ cuni 4900 dom cdm 5667 β cima 5670 Ord word 6354 Oncon0 6355 Lim wlim 6356 Fun wfun 6528 βcfv 6534 π 1cr1 9754 rankcrnk 9755 |
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 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-r1 9756 df-rank 9757 |
This theorem is referenced by: rankr1id 9854 ackbij2 10235 wunom 10712 r1limwun 10728 inar1 10767 r1tskina 10774 r1rankcld 43504 |
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