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Mirrors > Home > MPE Home > Th. List > sssucid | Structured version Visualization version 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 3738 | . 2 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝐴}) | |
2 | df-suc 5646 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
3 | 1, 2 | sseqtr4i 3601 | 1 ⊢ 𝐴 ⊆ suc 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3538 ⊆ wss 3540 {csn 4125 suc csuc 5642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-in 3547 df-ss 3554 df-suc 5646 |
This theorem is referenced by: trsuc 5727 suceloni 6905 limsssuc 6942 oaordi 7513 omeulem1 7549 oelim2 7562 nnaordi 7585 phplem4 8027 php 8029 onomeneq 8035 fiint 8122 cantnfval2 8449 cantnfle 8451 cantnfp1lem3 8460 cnfcomlem 8479 ranksuc 8611 fseqenlem1 8730 pwsdompw 8909 fin1a2lem12 9116 canthp1lem2 9354 nofulllem5 31105 limsucncmpi 31614 finxpreclem3 32406 clsk1independent 37364 suctrALT 38083 suctrALT2VD 38093 suctrALT2 38094 suctrALTcf 38180 suctrALTcfVD 38181 suctrALT3 38182 |
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