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Mirrors > Home > MPE Home > Th. List > frirr | Structured version Visualization version Unicode version |
Description: A well-founded relation is irreflexive. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 2-Jan-1994.) (Revised by Mario Carneiro, 22-Jun-2015.) |
Ref | Expression |
---|---|
frirr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 463 |
. . 3
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2 | simpr 467 |
. . . 4
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3 | 2 | snssd 4130 |
. . 3
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4 | snnzg 4102 |
. . . 4
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5 | 4 | adantl 472 |
. . 3
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6 | snex 4658 |
. . . 4
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7 | 6 | frc 4822 |
. . 3
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8 | 1, 3, 5, 7 | syl3anc 1276 |
. 2
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9 | rabeq0 3766 |
. . . . . 6
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10 | breq2 4422 |
. . . . . . . 8
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11 | 10 | notbid 300 |
. . . . . . 7
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12 | 11 | ralbidv 2839 |
. . . . . 6
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13 | 9, 12 | syl5bb 265 |
. . . . 5
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14 | 13 | rexsng 4019 |
. . . 4
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15 | breq1 4421 |
. . . . . 6
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16 | 15 | notbid 300 |
. . . . 5
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17 | 16 | ralsng 4018 |
. . . 4
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18 | 14, 17 | bitrd 261 |
. . 3
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19 | 18 | adantl 472 |
. 2
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20 | 8, 19 | mpbid 215 |
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 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-sep 4541 ax-nul 4550 ax-pr 4656 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-ral 2754 df-rex 2755 df-rab 2758 df-v 3059 df-sbc 3280 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-sn 3981 df-pr 3983 df-op 3987 df-br 4419 df-fr 4815 |
This theorem is referenced by: efrirr 4837 predfrirr 5432 dfwe2 6640 bnj1417 29900 efrunt 30390 ifr0 36848 |
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