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Mirrors > Home > MPE Home > Th. List > zfcndpow | Structured version Visualization version Unicode version |
Description: Axiom of Power Sets ax-pow 4594, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness," dtru 4607. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
zfcndpow |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dtru 4607 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | exnal 1709 |
. . . . 5
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3 | 1, 2 | mpbir 214 |
. . . 4
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4 | nfe1 1928 |
. . . . 5
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5 | axpownd 9051 |
. . . . 5
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6 | 4, 5 | exlimi 2005 |
. . . 4
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7 | 3, 6 | ax-mp 5 |
. . 3
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8 | 19.9v 1822 |
. . . . . . . 8
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9 | 19.3v 1823 |
. . . . . . . 8
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10 | 8, 9 | imbi12i 332 |
. . . . . . 7
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11 | 10 | albii 1701 |
. . . . . 6
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12 | 11 | imbi1i 331 |
. . . . 5
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13 | 12 | albii 1701 |
. . . 4
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14 | 13 | exbii 1728 |
. . 3
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15 | 7, 14 | mpbi 213 |
. 2
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16 | elequ1 1904 |
. . . . . . 7
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17 | elequ1 1904 |
. . . . . . 7
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18 | 16, 17 | imbi12d 326 |
. . . . . 6
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19 | 18 | cbvalv 2126 |
. . . . 5
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20 | 19 | imbi1i 331 |
. . . 4
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21 | 20 | albii 1701 |
. . 3
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22 | 21 | exbii 1728 |
. 2
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23 | 15, 22 | mpbir 214 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-8 1899 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-sep 4538 ax-nul 4547 ax-pow 4594 ax-pr 4652 ax-reg 8132 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-ral 2753 df-rex 2754 df-v 3058 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-nul 3743 df-pw 3964 df-sn 3980 df-pr 3982 |
This theorem is referenced by: (None) |
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