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| Mirrors > Home > MPE Home > Th. List > rankon | Structured version Visualization version GIF version | ||
| Description: The rank of a set is an ordinal number. Proposition 9.15(1) of [TakeutiZaring] p. 79. (Contributed by NM, 5-Oct-2003.) (Revised by Mario Carneiro, 12-Sep-2013.) |
| Ref | Expression |
|---|---|
| rankon | ⊢ (rank‘𝐴) ∈ On |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rankf 9834 | . 2 ⊢ rank:∪ (𝑅1 “ On)⟶On | |
| 2 | 0elon 6438 | . 2 ⊢ ∅ ∈ On | |
| 3 | 1, 2 | f0cli 7118 | 1 ⊢ (rank‘𝐴) ∈ On |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ∪ cuni 4907 “ cima 5688 Oncon0 6384 ‘cfv 6561 𝑅1cr1 9802 rankcrnk 9803 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-r1 9804 df-rank 9805 |
| This theorem is referenced by: rankr1ai 9838 rankr1bg 9843 rankr1clem 9860 rankr1c 9861 rankpwi 9863 rankelb 9864 wfelirr 9865 rankval3b 9866 ranksnb 9867 rankr1a 9876 bndrank 9881 unbndrank 9882 rankunb 9890 rankprb 9891 rankuni2b 9893 rankuni 9903 rankuniss 9906 rankval4 9907 rankbnd2 9909 rankc1 9910 rankc2 9911 rankelun 9912 rankelpr 9913 rankelop 9914 rankmapu 9918 rankxplim 9919 rankxplim3 9921 rankxpsuc 9922 tcrank 9924 scottex 9925 scott0 9926 dfac12lem2 10185 hsmexlem5 10470 r1limwun 10776 wunex3 10781 rankcf 10817 grur1 10860 elhf2 36176 hfuni 36185 dfac11 43074 gruex 44317 |
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