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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > predep | Unicode version |
Description: The predecessor under the epsilon relationship is equivalent to an intersection. (Contributed by Scott Fenton, 27-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
predep |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pred 25390 |
. 2
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2 | relcnv 5209 |
. . . . 5
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3 | relimasn 5194 |
. . . . 5
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4 | 2, 3 | ax-mp 8 |
. . . 4
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5 | vex 2927 |
. . . . . . 7
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6 | brcnvg 5020 |
. . . . . . 7
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7 | 5, 6 | mpan2 653 |
. . . . . 6
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8 | epelg 4463 |
. . . . . 6
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9 | 7, 8 | bitrd 245 |
. . . . 5
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10 | 9 | abbi1dv 2528 |
. . . 4
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11 | 4, 10 | syl5eq 2456 |
. . 3
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12 | 11 | ineq2d 3510 |
. 2
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13 | 1, 12 | syl5eq 2456 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: predon 25415 epsetlike 25416 omsinds 25441 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2393 ax-sep 4298 ax-nul 4306 ax-pr 4371 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2266 df-mo 2267 df-clab 2399 df-cleq 2405 df-clel 2408 df-nfc 2537 df-ne 2577 df-ral 2679 df-rex 2680 df-rab 2683 df-v 2926 df-sbc 3130 df-dif 3291 df-un 3293 df-in 3295 df-ss 3302 df-nul 3597 df-if 3708 df-sn 3788 df-pr 3789 df-op 3791 df-br 4181 df-opab 4235 df-eprel 4462 df-xp 4851 df-rel 4852 df-cnv 4853 df-dm 4855 df-rn 4856 df-res 4857 df-ima 4858 df-pred 25390 |
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