![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 9791 | . 2 ⊢ rank:∪ (𝑅1 “ On)⟶On | |
2 | 0elon 6412 | . 2 ⊢ ∅ ∈ On | |
3 | 1, 2 | f0cli 7093 | 1 ⊢ (rank‘𝐴) ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ∪ cuni 4902 “ cima 5672 Oncon0 6358 ‘cfv 6537 𝑅1cr1 9759 rankcrnk 9760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-r1 9761 df-rank 9762 |
This theorem is referenced by: rankr1ai 9795 rankr1bg 9800 rankr1clem 9817 rankr1c 9818 rankpwi 9820 rankelb 9821 wfelirr 9822 rankval3b 9823 ranksnb 9824 rankr1a 9833 bndrank 9838 unbndrank 9839 rankunb 9847 rankprb 9848 rankuni2b 9850 rankuni 9860 rankuniss 9863 rankval4 9864 rankbnd2 9866 rankc1 9867 rankc2 9868 rankelun 9869 rankelpr 9870 rankelop 9871 rankmapu 9875 rankxplim 9876 rankxplim3 9878 rankxpsuc 9879 tcrank 9881 scottex 9882 scott0 9883 dfac12lem2 10141 hsmexlem5 10427 r1limwun 10733 wunex3 10738 rankcf 10774 grur1 10817 elhf2 35680 hfuni 35689 dfac11 42387 gruex 43638 |
Copyright terms: Public domain | W3C validator |