![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-sucexg | GIF version |
Description: sucexg 4190 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-sucexg | ⊢ (A ∈ 𝑉 → suc A ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-snexg 9367 | . . . 4 ⊢ (A ∈ 𝑉 → {A} ∈ V) | |
2 | 1 | pm4.71i 371 | . . 3 ⊢ (A ∈ 𝑉 ↔ (A ∈ 𝑉 ∧ {A} ∈ V)) |
3 | 2 | biimpi 113 | . 2 ⊢ (A ∈ 𝑉 → (A ∈ 𝑉 ∧ {A} ∈ V)) |
4 | bj-unexg 9376 | . 2 ⊢ ((A ∈ 𝑉 ∧ {A} ∈ V) → (A ∪ {A}) ∈ V) | |
5 | df-suc 4074 | . . . 4 ⊢ suc A = (A ∪ {A}) | |
6 | 5 | eleq1i 2100 | . . 3 ⊢ (suc A ∈ V ↔ (A ∪ {A}) ∈ V) |
7 | 6 | biimpri 124 | . 2 ⊢ ((A ∪ {A}) ∈ V → suc A ∈ V) |
8 | 3, 4, 7 | 3syl 17 | 1 ⊢ (A ∈ 𝑉 → suc A ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1390 Vcvv 2551 ∪ cun 2909 {csn 3367 suc csuc 4068 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-pr 3935 ax-un 4136 ax-bd0 9268 ax-bdor 9271 ax-bdex 9274 ax-bdeq 9275 ax-bdel 9276 ax-bdsb 9277 ax-bdsep 9339 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rex 2306 df-v 2553 df-un 2916 df-sn 3373 df-pr 3374 df-uni 3572 df-suc 4074 df-bdc 9296 |
This theorem is referenced by: bj-sucex 9378 |
Copyright terms: Public domain | W3C validator |