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Mirrors > Home > ILE Home > Th. List > onin | Unicode version |
Description: The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.) |
Ref | Expression |
---|---|
onin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 4267 | . . 3 | |
2 | eloni 4267 | . . 3 | |
3 | ordin 4277 | . . 3 | |
4 | 1, 2, 3 | syl2an 287 | . 2 |
5 | simpl 108 | . . 3 | |
6 | inex1g 4034 | . . 3 | |
7 | elong 4265 | . . 3 | |
8 | 5, 6, 7 | 3syl 17 | . 2 |
9 | 4, 8 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 1465 cvv 2660 cin 3040 word 4254 con0 4255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-in 3047 df-ss 3054 df-uni 3707 df-tr 3997 df-iord 4258 df-on 4260 |
This theorem is referenced by: tfrlem5 6179 |
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