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Mirrors > Home > MPE Home > Th. List > sucexg | Structured version Visualization version GIF version |
Description: The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.) |
Ref | Expression |
---|---|
sucexg | ⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3510 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | sucexb 7513 | . 2 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | |
3 | 1, 2 | sylib 219 | 1 ⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3492 suc csuc 6186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-sn 4558 df-pr 4560 df-uni 4831 df-suc 6190 |
This theorem is referenced by: sucex 7515 suceloni 7517 hsmexlem1 9836 dfon2lem3 32927 inaex 40510 |
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