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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 7175 | . 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 2139 Vcvv 3340 suc csuc 5886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-rex 3056 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-sn 4322 df-pr 4324 df-uni 4589 df-suc 5890 |
This theorem is referenced by: orduninsuc 7208 tfindsg 7225 tfinds2 7228 finds 7257 findsg 7258 finds2 7259 seqomlem1 7714 oasuc 7773 onasuc 7777 infensuc 8303 phplem4 8307 php 8309 inf0 8691 inf3lem1 8698 dfom3 8717 cantnflt 8742 cantnflem1 8759 cnfcom 8770 infxpenlem 9026 pwsdompw 9218 ackbij1lem5 9238 cfslb2n 9282 cfsmolem 9284 fin1a2lem12 9425 axdc4lem 9469 alephreg 9596 bnj986 31331 bnj1018 31339 dfon2lem7 31999 bj-1ex 33244 bj-2ex 33245 cnfinltrel 33552 dford3lem2 38096 |
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