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Mirrors > Home > ILE Home > Th. List > elomssom | Unicode version |
Description: A natural number ordinal is, as a set, included in the set of natural number ordinals. (Contributed by NM, 21-Jun-1998.) Extract this result from the previous proof of elnn 4566. (Revised by BJ, 7-Aug-2024.) |
Ref | Expression |
---|---|
elomssom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 3151 | . 2 | |
2 | sseq1 3151 | . 2 | |
3 | sseq1 3151 | . 2 | |
4 | sseq1 3151 | . 2 | |
5 | 0ss 3432 | . 2 | |
6 | unss 3281 | . . . . 5 | |
7 | vex 2715 | . . . . . . 7 | |
8 | 7 | snss 3686 | . . . . . 6 |
9 | 8 | anbi2i 453 | . . . . 5 |
10 | df-suc 4332 | . . . . . 6 | |
11 | 10 | sseq1i 3154 | . . . . 5 |
12 | 6, 9, 11 | 3bitr4i 211 | . . . 4 |
13 | 12 | biimpi 119 | . . 3 |
14 | 13 | expcom 115 | . 2 |
15 | 1, 2, 3, 4, 5, 14 | finds 4560 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2128 cun 3100 wss 3102 c0 3394 csn 3560 csuc 4326 com 4550 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-iinf 4548 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-uni 3774 df-int 3809 df-suc 4332 df-iom 4551 |
This theorem is referenced by: elnn 4566 2ssom 13419 |
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