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Mirrors > Home > ILE Home > Th. List > axpweq | Unicode version |
Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4098 is not used by the proof. (Contributed by NM, 22-Jun-2009.) |
Ref | Expression |
---|---|
axpweq.1 |
Ref | Expression |
---|---|
axpweq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwidg 3524 | . . . 4 | |
2 | pweq 3513 | . . . . . 6 | |
3 | 2 | eleq2d 2209 | . . . . 5 |
4 | 3 | spcegv 2774 | . . . 4 |
5 | 1, 4 | mpd 13 | . . 3 |
6 | elex 2697 | . . . 4 | |
7 | 6 | exlimiv 1577 | . . 3 |
8 | 5, 7 | impbii 125 | . 2 |
9 | vex 2689 | . . . . 5 | |
10 | 9 | elpw2 4082 | . . . 4 |
11 | pwss 3526 | . . . . 5 | |
12 | dfss2 3086 | . . . . . . 7 | |
13 | 12 | imbi1i 237 | . . . . . 6 |
14 | 13 | albii 1446 | . . . . 5 |
15 | 11, 14 | bitri 183 | . . . 4 |
16 | 10, 15 | bitri 183 | . . 3 |
17 | 16 | exbii 1584 | . 2 |
18 | 8, 17 | bitri 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wal 1329 wceq 1331 wex 1468 wcel 1480 cvv 2686 wss 3071 cpw 3510 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-in 3077 df-ss 3084 df-pw 3512 |
This theorem is referenced by: (None) |
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