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| Mirrors > Home > MPE Home > Th. List > sucexb | Structured version Visualization version GIF version | ||
| Description: A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.) | 
| Ref | Expression | 
|---|---|
| sucexb | ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | unexb 7767 | . 2 ⊢ ((𝐴 ∈ V ∧ {𝐴} ∈ V) ↔ (𝐴 ∪ {𝐴}) ∈ V) | |
| 2 | snex 5436 | . . 3 ⊢ {𝐴} ∈ V | |
| 3 | 2 | biantru 529 | . 2 ⊢ (𝐴 ∈ V ↔ (𝐴 ∈ V ∧ {𝐴} ∈ V)) | 
| 4 | df-suc 6390 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
| 5 | 4 | eleq1i 2832 | . 2 ⊢ (suc 𝐴 ∈ V ↔ (𝐴 ∪ {𝐴}) ∈ V) | 
| 6 | 1, 3, 5 | 3bitr4i 303 | 1 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 {csn 4626 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: sucexg 7825 onsucb 7837 ordsucelsuc 7842 oeordi 8625 suc11reg 9659 rankxpsuc 9922 isf32lem2 10394 limsucncmpi 36446 | 
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