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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 9207 | . 2 ⊢ rank:∪ (𝑅1 “ On)⟶On | |
2 | 0elon 6212 | . 2 ⊢ ∅ ∈ On | |
3 | 1, 2 | f0cli 6841 | 1 ⊢ (rank‘𝐴) ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 ∪ cuni 4800 “ cima 5522 Oncon0 6159 ‘cfv 6324 𝑅1cr1 9175 rankcrnk 9176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-r1 9177 df-rank 9178 |
This theorem is referenced by: rankr1ai 9211 rankr1bg 9216 rankr1clem 9233 rankr1c 9234 rankpwi 9236 rankelb 9237 wfelirr 9238 rankval3b 9239 ranksnb 9240 rankr1a 9249 bndrank 9254 unbndrank 9255 rankunb 9263 rankprb 9264 rankuni2b 9266 rankuni 9276 rankuniss 9279 rankval4 9280 rankbnd2 9282 rankc1 9283 rankc2 9284 rankelun 9285 rankelpr 9286 rankelop 9287 rankmapu 9291 rankxplim 9292 rankxplim3 9294 rankxpsuc 9295 tcrank 9297 scottex 9298 scott0 9299 dfac12lem2 9555 hsmexlem5 9841 r1limwun 10147 wunex3 10152 rankcf 10188 grur1 10231 elhf2 33749 hfuni 33758 dfac11 40006 gruex 41006 |
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