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Mirrors > Home > ILE Home > Th. List > sucex | 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 4374 | . 2 ⊢ (𝐴 ∈ V → suc 𝐴 ∈ V) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ suc 𝐴 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1463 Vcvv 2657 suc csuc 4247 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-rex 2396 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-uni 3703 df-suc 4253 |
This theorem is referenced by: finds 4474 finds2 4475 limom 4487 tfrexlem 6185 oafnex 6294 sucinc 6295 oacl 6310 nnsucelsuc 6341 phplem4 6702 php5 6705 phpm 6712 |
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