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Mirrors > Home > MPE Home > Th. List > sucprcreg | Structured version Visualization version Unicode version |
Description: A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity). (Contributed by NM, 9-Jul-2004.) (Proof shortened by BJ, 16-Apr-2019.) |
Ref | Expression |
---|---|
sucprcreg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sucprc 5498 |
. 2
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2 | elirr 8113 |
. . . 4
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3 | df-suc 5429 |
. . . . . . . 8
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4 | 3 | eqeq1i 2456 |
. . . . . . 7
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5 | ssequn2 3607 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 4, 5 | bitr4i 256 |
. . . . . 6
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7 | 6 | biimpi 198 |
. . . . 5
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8 | snidg 3994 |
. . . . 5
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9 | ssel2 3427 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | 7, 8, 9 | syl2an 480 |
. . . 4
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11 | 2, 10 | mto 180 |
. . 3
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12 | 11 | imnani 425 |
. 2
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13 | 1, 12 | impbii 191 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pr 4639 ax-reg 8107 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-v 3047 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-sn 3969 df-pr 3971 df-suc 5429 |
This theorem is referenced by: (None) |
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