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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 7790 | . 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 2107 Vcvv 3475 suc csuc 6364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-sn 4629 df-pr 4631 df-uni 4909 df-suc 6368 |
This theorem is referenced by: orduninsuc 7829 tfindsg 7847 tfinds2 7850 finds 7886 findsg 7887 finds2 7888 seqomlem1 8447 2oexOLD 8484 oasuc 8521 onasuc 8525 naddcllem 8672 infensuc 9152 phplem4OLD 9217 phpOLD 9219 inf0 9613 inf3lem1 9620 dfom3 9639 cantnflt 9664 cantnflem1 9681 cnfcom 9692 brttrcl2 9706 ssttrcl 9707 ttrcltr 9708 ttrclss 9712 ttrclselem2 9718 infxpenlem 10005 pwsdompw 10196 cfslb2n 10260 cfsmolem 10262 fin1a2lem12 10403 axdc4lem 10447 alephreg 10574 bnj986 33955 bnj1018g 33963 bnj1018 33964 satf 34333 dfon2lem7 34750 rdgssun 36248 dford3lem2 41752 |
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