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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 9217 | . 2 ⊢ rank:∪ (𝑅1 “ On)⟶On | |
2 | 0elon 6238 | . 2 ⊢ ∅ ∈ On | |
3 | 1, 2 | f0cli 6858 | 1 ⊢ (rank‘𝐴) ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 ∪ cuni 4831 “ cima 5552 Oncon0 6185 ‘cfv 6349 𝑅1cr1 9185 rankcrnk 9186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 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 3496 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 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 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 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-r1 9187 df-rank 9188 |
This theorem is referenced by: rankr1ai 9221 rankr1bg 9226 rankr1clem 9243 rankr1c 9244 rankpwi 9246 rankelb 9247 wfelirr 9248 rankval3b 9249 ranksnb 9250 rankr1a 9259 bndrank 9264 unbndrank 9265 rankunb 9273 rankprb 9274 rankuni2b 9276 rankuni 9286 rankuniss 9289 rankval4 9290 rankbnd2 9292 rankc1 9293 rankc2 9294 rankelun 9295 rankelpr 9296 rankelop 9297 rankmapu 9301 rankxplim 9302 rankxplim3 9304 rankxpsuc 9305 tcrank 9307 scottex 9308 scott0 9309 dfac12lem2 9564 hsmexlem5 9846 r1limwun 10152 wunex3 10157 rankcf 10193 grur1 10236 elhf2 33631 hfuni 33640 dfac11 39655 gruex 40627 |
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