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Mirrors > Home > MPE Home > Th. List > pwuninel | Structured version Unicode version |
Description: The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. See also pwuninel2 6893. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
pwuninel |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3077 |
. . . 4
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2 | pwexb 6487 |
. . . 4
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3 | 1, 2 | sylibr 212 |
. . 3
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4 | pwuninel2 6893 |
. . 3
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5 | 3, 4 | syl 16 |
. 2
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6 | id 22 |
. 2
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7 | 5, 6 | pm2.61i 164 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4511 ax-nul 4519 ax-pow 4568 ax-pr 4629 ax-un 6472 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-rex 2801 df-rab 2804 df-v 3070 df-dif 3429 df-un 3431 df-in 3433 df-ss 3440 df-nul 3736 df-pw 3960 df-sn 3976 df-pr 3978 df-uni 4190 |
This theorem is referenced by: undefnel2 6896 disjen 7568 pnfnre 9526 kelac2lem 29555 kelac2 29556 |
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