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Mirrors > Home > MPE Home > Th. List > rankr1ag | Structured version Visualization version GIF version |
Description: A version of rankr1a 9835 that is suitable without assuming Regularity or Replacement. (Contributed by Mario Carneiro, 3-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
rankr1ag | β’ ((π΄ β βͺ (π 1 β On) β§ π΅ β dom π 1) β (π΄ β (π 1βπ΅) β (rankβπ΄) β π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankr1ai 9797 | . 2 β’ (π΄ β (π 1βπ΅) β (rankβπ΄) β π΅) | |
2 | r1funlim 9765 | . . . . . . . 8 β’ (Fun π 1 β§ Lim dom π 1) | |
3 | 2 | simpri 485 | . . . . . . 7 β’ Lim dom π 1 |
4 | limord 6424 | . . . . . . 7 β’ (Lim dom π 1 β Ord dom π 1) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 β’ Ord dom π 1 |
6 | ordelord 6386 | . . . . . 6 β’ ((Ord dom π 1 β§ π΅ β dom π 1) β Ord π΅) | |
7 | 5, 6 | mpan 687 | . . . . 5 β’ (π΅ β dom π 1 β Ord π΅) |
8 | 7 | adantl 481 | . . . 4 β’ ((π΄ β βͺ (π 1 β On) β§ π΅ β dom π 1) β Ord π΅) |
9 | ordsucss 7810 | . . . 4 β’ (Ord π΅ β ((rankβπ΄) β π΅ β suc (rankβπ΄) β π΅)) | |
10 | 8, 9 | syl 17 | . . 3 β’ ((π΄ β βͺ (π 1 β On) β§ π΅ β dom π 1) β ((rankβπ΄) β π΅ β suc (rankβπ΄) β π΅)) |
11 | rankidb 9799 | . . . . 5 β’ (π΄ β βͺ (π 1 β On) β π΄ β (π 1βsuc (rankβπ΄))) | |
12 | elfvdm 6928 | . . . . 5 β’ (π΄ β (π 1βsuc (rankβπ΄)) β suc (rankβπ΄) β dom π 1) | |
13 | 11, 12 | syl 17 | . . . 4 β’ (π΄ β βͺ (π 1 β On) β suc (rankβπ΄) β dom π 1) |
14 | r1ord3g 9778 | . . . 4 β’ ((suc (rankβπ΄) β dom π 1 β§ π΅ β dom π 1) β (suc (rankβπ΄) β π΅ β (π 1βsuc (rankβπ΄)) β (π 1βπ΅))) | |
15 | 13, 14 | sylan 579 | . . 3 β’ ((π΄ β βͺ (π 1 β On) β§ π΅ β dom π 1) β (suc (rankβπ΄) β π΅ β (π 1βsuc (rankβπ΄)) β (π 1βπ΅))) |
16 | 11 | adantr 480 | . . . 4 β’ ((π΄ β βͺ (π 1 β On) β§ π΅ β dom π 1) β π΄ β (π 1βsuc (rankβπ΄))) |
17 | ssel 3975 | . . . 4 β’ ((π 1βsuc (rankβπ΄)) β (π 1βπ΅) β (π΄ β (π 1βsuc (rankβπ΄)) β π΄ β (π 1βπ΅))) | |
18 | 16, 17 | syl5com 31 | . . 3 β’ ((π΄ β βͺ (π 1 β On) β§ π΅ β dom π 1) β ((π 1βsuc (rankβπ΄)) β (π 1βπ΅) β π΄ β (π 1βπ΅))) |
19 | 10, 15, 18 | 3syld 60 | . 2 β’ ((π΄ β βͺ (π 1 β On) β§ π΅ β dom π 1) β ((rankβπ΄) β π΅ β π΄ β (π 1βπ΅))) |
20 | 1, 19 | impbid2 225 | 1 β’ ((π΄ β βͺ (π 1 β On) β§ π΅ β dom π 1) β (π΄ β (π 1βπ΅) β (rankβπ΄) β π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β wcel 2105 β wss 3948 βͺ cuni 4908 dom cdm 5676 β cima 5679 Ord word 6363 Oncon0 6364 Lim wlim 6365 suc csuc 6366 Fun wfun 6537 βcfv 6543 π 1cr1 9761 rankcrnk 9762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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-reu 3376 df-rab 3432 df-v 3475 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-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-lim 6369 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-ov 7415 df-om 7860 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-r1 9763 df-rank 9764 |
This theorem is referenced by: rankr1bg 9802 rankr1clem 9819 rankr1c 9820 rankval3b 9825 onssr1 9830 r1pw 9844 r1pwcl 9846 hsmexlem6 10430 r1limwun 10735 inatsk 10777 grur1 10819 |
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