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Mirrors > Home > MPE Home > Th. List > smoel2 | Structured version Visualization version Unicode version |
Description: A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Ref | Expression |
---|---|
smoel2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fndm 5656 |
. . . . . 6
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2 | 1 | eleq2d 2514 |
. . . . 5
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3 | 2 | anbi1d 716 |
. . . 4
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4 | 3 | biimprd 231 |
. . 3
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5 | smoel 7065 |
. . . 4
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6 | 5 | 3expib 1213 |
. . 3
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7 | 4, 6 | sylan9 667 |
. 2
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8 | 7 | 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 1672 ax-4 1685 ax-5 1761 ax-6 1808 ax-7 1854 ax-10 1918 ax-11 1923 ax-12 1936 ax-13 2091 ax-ext 2431 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 988 df-tru 1450 df-ex 1667 df-nf 1671 df-sb 1801 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ral 2741 df-rex 2742 df-rab 2745 df-v 3014 df-dif 3374 df-un 3376 df-in 3378 df-ss 3385 df-nul 3699 df-if 3849 df-sn 3936 df-pr 3938 df-op 3942 df-uni 4168 df-br 4374 df-tr 4469 df-ord 5404 df-iota 5524 df-fn 5563 df-fv 5568 df-smo 7051 |
This theorem is referenced by: smo11 7069 smoord 7070 smogt 7072 cofsmo 8685 |
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