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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 4590. (Revised by BJ, 7-Aug-2024.) |
Ref | Expression |
---|---|
elomssom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 3170 | . 2 | |
2 | sseq1 3170 | . 2 | |
3 | sseq1 3170 | . 2 | |
4 | sseq1 3170 | . 2 | |
5 | 0ss 3453 | . 2 | |
6 | unss 3301 | . . . . 5 | |
7 | vex 2733 | . . . . . . 7 | |
8 | 7 | snss 3709 | . . . . . 6 |
9 | 8 | anbi2i 454 | . . . . 5 |
10 | df-suc 4356 | . . . . . 6 | |
11 | 10 | sseq1i 3173 | . . . . 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 4584 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2141 cun 3119 wss 3121 c0 3414 csn 3583 csuc 4350 com 4574 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-int 3832 df-suc 4356 df-iom 4575 |
This theorem is referenced by: elnn 4590 2ssom 6503 nninfwlpoimlemginf 7152 |
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