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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 9710 | . . . . . . . . . 10 β’ (Fun π 1 β§ Lim dom π 1) | |
2 | 1 | simpri 487 | . . . . . . . . 9 β’ Lim dom π 1 |
3 | limord 6381 | . . . . . . . . 9 β’ (Lim dom π 1 β Ord dom π 1) | |
4 | ordtr1 6364 | . . . . . . . . 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 460 | . . . . . . 7 β’ ((π΄ β dom π 1 β§ π₯ β π΄) β π₯ β dom π 1) |
7 | rankonidlem 9772 | . . . . . . 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 497 | . . . . 5 β’ ((π΄ β dom π 1 β§ π₯ β π΄) β (rankβπ₯) = π₯) |
10 | simpr 486 | . . . . 5 β’ ((π΄ β dom π 1 β§ π₯ β π΄) β π₯ β π΄) | |
11 | 9, 10 | eqeltrd 2834 | . . . 4 β’ ((π΄ β dom π 1 β§ π₯ β π΄) β (rankβπ₯) β π΄) |
12 | 8 | simpld 496 | . . . . 5 β’ ((π΄ β dom π 1 β§ π₯ β π΄) β π₯ β βͺ (π 1 β On)) |
13 | simpl 484 | . . . . 5 β’ ((π΄ β dom π 1 β§ π₯ β π΄) β π΄ β dom π 1) | |
14 | rankr1ag 9746 | . . . . 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 585 | . . . 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 414 | . 2 β’ (π΄ β dom π 1 β (π₯ β π΄ β π₯ β (π 1βπ΄))) |
18 | 17 | ssrdv 3954 | 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 397 = wceq 1542 β wcel 2107 β wss 3914 βͺ cuni 4869 dom cdm 5637 β cima 5640 Ord word 6320 Oncon0 6321 Lim wlim 6322 Fun wfun 6494 βcfv 6500 π 1cr1 9706 rankcrnk 9707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-r1 9708 df-rank 9709 |
This theorem is referenced by: rankr1id 9806 ackbij2 10187 wunom 10664 r1limwun 10680 inar1 10719 r1tskina 10726 r1rankcld 42603 |
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