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Mirrors > Home > MPE Home > Th. List > oninton | Structured version Visualization version Unicode version |
Description: The intersection of a nonempty collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by NM, 29-Jan-1997.) |
Ref | Expression |
---|---|
oninton |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onint 6995 | . . . 4 | |
2 | 1 | ex 450 | . . 3 |
3 | ssel 3597 | . . 3 | |
4 | 2, 3 | syld 47 | . 2 |
5 | 4 | imp 445 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wcel 1990 wne 2794 wss 3574 c0 3915 cint 4475 con0 5723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 |
This theorem is referenced by: onintrab 7001 onnmin 7003 onminex 7007 onmindif2 7012 iinon 7437 oawordeulem 7634 nnawordex 7717 tz9.12lem1 8650 rankf 8657 cardf2 8769 cff 9070 coftr 9095 sltval2 31809 nocvxminlem 31893 dnnumch3lem 37616 dnnumch3 37617 |
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