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Mirrors > Home > ILE Home > Th. List > 0elpw | Unicode version |
Description: Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.) |
Ref | Expression |
---|---|
0elpw |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 3452 | . 2 | |
2 | 0ex 4114 | . . 3 | |
3 | 2 | elpw 3570 | . 2 |
4 | 1, 3 | mpbir 145 | 1 |
Colors of variables: wff set class |
Syntax hints: wcel 2141 wss 3121 c0 3414 cpw 3564 |
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-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-nul 4113 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 |
This theorem is referenced by: ordpwsucexmid 4552 pw1on 7192 pw1ne0 7194 pw1nct 13998 |
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