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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 9212 | . 2 ⊢ rank:∪ (𝑅1 “ On)⟶On | |
2 | 0elon 6238 | . 2 ⊢ ∅ ∈ On | |
3 | 1, 2 | f0cli 6857 | 1 ⊢ (rank‘𝐴) ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ∪ cuni 4832 “ cima 5552 Oncon0 6185 ‘cfv 6349 𝑅1cr1 9180 rankcrnk 9181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-om 7569 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-r1 9182 df-rank 9183 |
This theorem is referenced by: rankr1ai 9216 rankr1bg 9221 rankr1clem 9238 rankr1c 9239 rankpwi 9241 rankelb 9242 wfelirr 9243 rankval3b 9244 ranksnb 9245 rankr1a 9254 bndrank 9259 unbndrank 9260 rankunb 9268 rankprb 9269 rankuni2b 9271 rankuni 9281 rankuniss 9284 rankval4 9285 rankbnd2 9287 rankc1 9288 rankc2 9289 rankelun 9290 rankelpr 9291 rankelop 9292 rankmapu 9296 rankxplim 9297 rankxplim3 9299 rankxpsuc 9300 tcrank 9302 scottex 9303 scott0 9304 dfac12lem2 9559 hsmexlem5 9841 r1limwun 10147 wunex3 10152 rankcf 10188 grur1 10231 elhf2 33534 hfuni 33543 dfac11 39542 gruex 40514 |
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