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| Description: The minimum of a class of ordinal numbers is less than the minimum of that class with its minimum removed. |
| Ref | Expression |
|---|---|
| onmindif2 |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | onnmin 3072 |
. . . . . . . . . 10
| |
| 2 | 1 | adantlr 402 |
. . . . . . . . 9
|
| 3 | ontri1 3038 |
. . . . . . . . . . 11
| |
| 4 | onsseleq 3056 |
. . . . . . . . . . 11
| |
| 5 | 3, 4 | bitr3d 541 |
. . . . . . . . . 10
|
| 6 | oninton 3069 |
. . . . . . . . . . 11
| |
| 7 | 6 | adantr 398 |
. . . . . . . . . 10
|
| 8 | ssel2 2115 |
. . . . . . . . . . 11
| |
| 9 | 8 | adantlr 402 |
. . . . . . . . . 10
|
| 10 | 5, 7, 9 | sylanc 482 |
. . . . . . . . 9
|
| 11 | 2, 10 | mpbid 202 |
. . . . . . . 8
|
| 12 | 11 | ord 239 |
. . . . . . 7
|
| 13 | eqcom 1524 |
. . . . . . 7
| |
| 14 | 12, 13 | syl6ib 219 |
. . . . . 6
|
| 15 | 14 | necon1ad 1678 |
. . . . 5
|
| 16 | 15 | expimpd 382 |
. . . 4
|
| 17 | eldifsn 2516 |
. . . 4
| |
| 18 | 16, 17 | syl5ib 213 |
. . 3
|
| 19 | 18 | r19.21aiv 1760 |
. 2
|
| 20 | intex 2784 |
. . . 4
| |
| 21 | elintg 2595 |
. . . 4
| |
| 22 | 20, 21 | sylbi 206 |
. . 3
|
| 23 | 22 | adantl 397 |
. 2
|
| 24 | 19, 23 | mpbird 203 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 1003 ax-gen 1004 ax-8 1005 ax-10 1007 ax-11 1008 ax-12 1009 ax-13 1010 ax-14 1011 ax-17 1012 ax-4 1014 ax-5o 1016 ax-6o 1019 ax-9o 1164 ax-10o 1182 ax-16 1252 ax-11o 1260 ax-ext 1504 ax-sep 2758 ax-pow 2798 ax-pr 2835 ax-un 2922 |
| This theorem depends on definitions: df-bi 154 df-or 231 df-an 232 df-3or 788 df-3an 789 df-ex 1022 df-sb 1214 df-eu 1424 df-mo 1425 df-clab 1510 df-cleq 1515 df-clel 1518 df-ne 1634 df-ral 1696 df-rex 1697 df-v 1859 df-dif 2100 df-un 2101 df-in 2102 df-ss 2104 df-nul 2332 df-pw 2454 df-sn 2464 df-pr 2465 df-tp 2467 df-op 2468 df-uni 2558 df-int 2588 df-br 2675 df-opab 2722 df-tr 2736 df-eprel 2888 df-po 2896 df-so 2906 df-fr 2974 df-we 2991 df-ord 3008 df-on 3009 |