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Mirrors > Home > MPE Home > Th. List > suc11 | Structured version Visualization version Unicode version |
Description: The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.) |
Ref | Expression |
---|---|
suc11 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 5411 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | ordn2lp 5421 |
. . . . . 6
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3 | ianor 495 |
. . . . . 6
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4 | 2, 3 | sylib 201 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | 1, 4 | syl 17 |
. . . 4
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6 | 5 | adantr 471 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | eqimss 3451 |
. . . . . 6
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8 | sucssel 5493 |
. . . . . 6
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9 | 7, 8 | syl5 33 |
. . . . 5
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10 | elsuci 5467 |
. . . . . . 7
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11 | 10 | ord 383 |
. . . . . 6
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12 | 11 | com12 32 |
. . . . 5
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13 | 9, 12 | syl9 73 |
. . . 4
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14 | eqimss2 3452 |
. . . . . 6
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15 | sucssel 5493 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 14, 15 | syl5 33 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | elsuci 5467 |
. . . . . . . 8
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18 | 17 | ord 383 |
. . . . . . 7
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19 | 18 | com12 32 |
. . . . . 6
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20 | eqcom 2458 |
. . . . . 6
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21 | 19, 20 | syl6ib 234 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 16, 21 | syl9 73 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 13, 22 | jaao 516 |
. . 3
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24 | 6, 23 | mpd 15 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | suceq 5466 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
26 | 24, 25 | impbid1 208 |
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-9 1899 ax-10 1918 ax-11 1923 ax-12 1936 ax-13 2091 ax-ext 2431 ax-sep 4496 ax-nul 4505 ax-pr 4611 |
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-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2623 df-ral 2741 df-rex 2742 df-rab 2745 df-v 3014 df-sbc 3235 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-opab 4433 df-tr 4469 df-eprel 4722 df-po 4732 df-so 4733 df-fr 4770 df-we 4772 df-ord 5404 df-on 5405 df-suc 5407 |
This theorem is referenced by: peano4 6702 limenpsi 7733 fin1a2lem2 8817 bnj168 29543 sltval2 30548 sltsolem1 30562 onsuct0 31106 |
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