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| Mirrors > Home > MPE Home > Th. List > sucex | Structured version Visualization version GIF version | ||
| Description: The successor of a set is a set. (Contributed by NM, 30-Aug-1993.) |
| Ref | Expression |
|---|---|
| sucex.1 | ⊢ 𝐴 ∈ V |
| Ref | Expression |
|---|---|
| sucex | ⊢ suc 𝐴 ∈ V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sucex.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | sucexg 7825 | . 2 ⊢ (𝐴 ∈ V → suc 𝐴 ∈ V) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ suc 𝐴 ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 suc csuc 6386 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-sn 4627 df-pr 4629 df-uni 4908 df-suc 6390 |
| This theorem is referenced by: orduninsuc 7864 tfindsg 7882 tfinds2 7885 finds 7918 findsg 7919 finds2 7920 seqomlem1 8490 oasuc 8562 onasuc 8566 naddcllem 8714 infensuc 9195 phplem4OLD 9257 phpOLD 9259 inf0 9661 inf3lem1 9668 dfom3 9687 cantnflt 9712 cantnflem1 9729 cnfcom 9740 brttrcl2 9754 ssttrcl 9755 ttrcltr 9756 ttrclss 9760 ttrclselem2 9766 infxpenlem 10053 pwsdompw 10243 cfslb2n 10308 cfsmolem 10310 fin1a2lem12 10451 axdc4lem 10495 alephreg 10622 bnj986 34969 bnj1018g 34977 bnj1018 34978 satf 35358 dfon2lem7 35790 rdgssun 37379 dford3lem2 43039 |
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