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Mirrors > Home > MPE Home > Th. List > trsuc | Structured version Visualization version Unicode version |
Description: A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
trsuc |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sssucid 5523 |
. . . . . 6
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2 | ssexg 4565 |
. . . . . 6
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3 | 1, 2 | mpan 681 |
. . . . 5
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4 | sucidg 5524 |
. . . . 5
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5 | 3, 4 | syl 17 |
. . . 4
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6 | 5 | ancri 559 |
. . 3
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7 | trel 4520 |
. . 3
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8 | 6, 7 | syl5 33 |
. 2
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9 | 8 | imp 435 |
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-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-sep 4541 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 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-v 3059 df-un 3421 df-in 3423 df-ss 3430 df-sn 3981 df-uni 4213 df-tr 4514 df-suc 5452 |
This theorem is referenced by: onuninsuci 6699 limsuc 6708 tz7.44-2 7156 cantnflt 8208 cantnfp1lem3 8216 cantnflem1b 8222 cantnflem1 8225 cnfcom 8236 axdc3lem2 8912 inar1 9231 bnj967 29806 limsuc2 35945 |
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