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Mirrors > Home > ILE Home > Th. List > sssucid | GIF version |
Description: A class is included in its own successor. Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized to arbitrary classes). (Contributed by NM, 31-May-1994.) |
Ref | Expression |
---|---|
sssucid | ⊢ 𝐴 ⊆ suc 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 3285 | . 2 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝐴}) | |
2 | df-suc 4349 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
3 | 1, 2 | sseqtrri 3177 | 1 ⊢ 𝐴 ⊆ suc 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∪ cun 3114 ⊆ wss 3116 {csn 3576 suc csuc 4343 |
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 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-ext 2147 |
This theorem depends on definitions: df-bi 116 df-tru 1346 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-v 2728 df-un 3120 df-in 3122 df-ss 3129 df-suc 4349 |
This theorem is referenced by: trsuc 4400 ordsuc 4540 0elnn 4596 sucinc 6413 sucinc2 6414 oasuc 6432 phplem4 6821 phplem4dom 6828 phplem4on 6833 fiintim 6894 fidcenumlemrk 6919 fidcenumlemr 6920 bj-nntrans 13833 |
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