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Mirrors > Home > MPE Home > Th. List > onelon | Structured version Visualization version GIF version |
Description: An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.) |
Ref | Expression |
---|---|
onelon | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 5874 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | ordelon 5888 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) | |
3 | 1, 2 | sylan 569 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∈ wcel 2145 Ord word 5863 Oncon0 5864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-tr 4887 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-ord 5867 df-on 5868 |
This theorem is referenced by: oneli 5976 ssorduni 7132 unon 7178 tfindsg2 7208 dfom2 7214 ordom 7221 onfununi 7591 onnseq 7594 dfrecs3 7622 tz7.48-2 7690 tz7.49 7693 oalim 7766 omlim 7767 oelim 7768 oaordi 7780 oalimcl 7794 oaass 7795 omordi 7800 omlimcl 7812 odi 7813 omass 7814 omeulem1 7816 omeulem2 7817 omopth2 7818 oewordri 7826 oeordsuc 7828 oelimcl 7834 oeeui 7836 oaabs2 7879 omabs 7881 omxpenlem 8217 hartogs 8605 card2on 8615 cantnfle 8732 cantnflt 8733 cantnfp1lem2 8740 cantnfp1lem3 8741 cantnfp1 8742 oemapvali 8745 cantnflem1b 8747 cantnflem1c 8748 cantnflem1d 8749 cantnflem1 8750 cantnflem2 8751 cantnflem3 8752 cantnflem4 8753 cantnf 8754 cnfcomlem 8760 cnfcom3lem 8764 cnfcom3 8765 r1ordg 8805 r1val3 8865 tskwe 8976 iscard 9001 cardmin2 9024 infxpenlem 9036 infxpenc2lem2 9043 alephordi 9097 alephord2i 9100 alephle 9111 cardaleph 9112 cfub 9273 cfsmolem 9294 zorn2lem5 9524 zorn2lem6 9525 ttukeylem6 9538 ttukeylem7 9539 ondomon 9587 cardmin 9588 alephval2 9596 alephreg 9606 smobeth 9610 winainflem 9717 inar1 9799 inatsk 9802 dfrdg2 32033 sltval2 32142 sltres 32148 nosepeq 32168 nosupno 32182 nosupres 32186 nosupbnd1lem1 32187 nosupbnd2lem1 32194 nosupbnd2 32195 dfrdg4 32391 ontopbas 32760 onpsstopbas 32762 onint1 32781 |
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