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Mirrors > Home > ILE Home > Th. List > sucexb | 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 4358 | . 2 ⊢ ((𝐴 ∈ V ∧ {𝐴} ∈ V) ↔ (𝐴 ∪ {𝐴}) ∈ V) | |
2 | snexg 4103 | . . 3 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
3 | 2 | pm4.71i 388 | . 2 ⊢ (𝐴 ∈ V ↔ (𝐴 ∈ V ∧ {𝐴} ∈ V)) |
4 | df-suc 4288 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
5 | 4 | eleq1i 2203 | . 2 ⊢ (suc 𝐴 ∈ V ↔ (𝐴 ∪ {𝐴}) ∈ V) |
6 | 1, 3, 5 | 3bitr4i 211 | 1 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ wcel 1480 Vcvv 2681 ∪ cun 3064 {csn 3522 suc csuc 4282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-uni 3732 df-suc 4288 |
This theorem is referenced by: sucexg 4409 sucelon 4414 onsucelsucr 4419 sucunielr 4421 peano2b 4523 |
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