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Mirrors > Home > MPE Home > Th. List > elnpi | Structured version Unicode version |
Description: Membership in positive reals. (Contributed by Mario Carneiro, 11-May-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elnpi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3081 |
. 2
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2 | simpl1 991 |
. 2
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3 | psseq2 3547 |
. . . . . 6
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4 | psseq1 3546 |
. . . . . 6
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5 | 3, 4 | anbi12d 710 |
. . . . 5
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6 | eleq2 2525 |
. . . . . . . . 9
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7 | 6 | imbi2d 316 |
. . . . . . . 8
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8 | 7 | albidv 1680 |
. . . . . . 7
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9 | rexeq 3018 |
. . . . . . 7
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10 | 8, 9 | anbi12d 710 |
. . . . . 6
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11 | 10 | raleqbi1dv 3025 |
. . . . 5
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12 | 5, 11 | anbi12d 710 |
. . . 4
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13 | df-np 9256 |
. . . 4
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14 | 12, 13 | elab2g 3209 |
. . 3
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15 | id 22 |
. . . . . 6
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16 | 15 | 3expib 1191 |
. . . . 5
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17 | 3simpc 987 |
. . . . 5
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18 | 16, 17 | impbid1 203 |
. . . 4
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19 | 18 | anbi1d 704 |
. . 3
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20 | 14, 19 | bitrd 253 |
. 2
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21 | 1, 2, 20 | pm5.21nii 353 |
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-10 1777 ax-11 1782 ax-12 1794 ax-13 1954 ax-ext 2431 |
This theorem depends on definitions: df-bi 185 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2602 df-ne 2647 df-ral 2801 df-rex 2802 df-v 3074 df-in 3438 df-ss 3445 df-pss 3447 df-np 9256 |
This theorem is referenced by: prn0 9264 prpssnq 9265 prcdnq 9268 prnmax 9270 |
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