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Mirrors > Home > MPE Home > Th. List > unblem1 | Structured version Unicode version |
Description: Lemma for unbnn 7678. After removing the successor of an element from an unbounded set of natural numbers, the intersection of the result belongs to the original unbounded set. (Contributed by NM, 3-Dec-2003.) |
Ref | Expression |
---|---|
unblem1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omsson 6589 |
. . . . . 6
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2 | sstr 3471 |
. . . . . 6
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3 | 1, 2 | mpan2 671 |
. . . . 5
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4 | 3 | ssdifssd 3601 |
. . . 4
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5 | 4 | ad2antrr 725 |
. . 3
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6 | ssel 3457 |
. . . . . 6
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7 | peano2b 6601 |
. . . . . 6
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8 | 6, 7 | syl6ib 226 |
. . . . 5
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9 | eleq1 2526 |
. . . . . . . 8
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10 | 9 | rexbidv 2864 |
. . . . . . 7
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11 | 10 | rspccva 3176 |
. . . . . 6
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12 | ssel 3457 |
. . . . . . . . . . 11
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13 | nnord 6593 |
. . . . . . . . . . . 12
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14 | ordn2lp 4846 |
. . . . . . . . . . . . . 14
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15 | imnan 422 |
. . . . . . . . . . . . . 14
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16 | 14, 15 | sylibr 212 |
. . . . . . . . . . . . 13
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17 | 16 | con2d 115 |
. . . . . . . . . . . 12
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18 | 13, 17 | syl 16 |
. . . . . . . . . . 11
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19 | 12, 18 | syl6 33 |
. . . . . . . . . 10
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20 | 19 | imdistand 692 |
. . . . . . . . 9
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21 | eldif 3445 |
. . . . . . . . . 10
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22 | ne0i 3750 |
. . . . . . . . . 10
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23 | 21, 22 | sylbir 213 |
. . . . . . . . 9
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24 | 20, 23 | syl6 33 |
. . . . . . . 8
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25 | 24 | expd 436 |
. . . . . . 7
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26 | 25 | rexlimdv 2944 |
. . . . . 6
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27 | 11, 26 | syl5 32 |
. . . . 5
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28 | 8, 27 | sylan2d 482 |
. . . 4
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29 | 28 | impl 620 |
. . 3
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30 | onint 6515 |
. . 3
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31 | 5, 29, 30 | syl2anc 661 |
. 2
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32 | 31 | eldifad 3447 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4520 ax-nul 4528 ax-pr 4638 ax-un 6481 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2649 df-ral 2803 df-rex 2804 df-rab 2807 df-v 3078 df-sbc 3293 df-dif 3438 df-un 3440 df-in 3442 df-ss 3449 df-pss 3451 df-nul 3745 df-if 3899 df-pw 3969 df-sn 3985 df-pr 3987 df-tp 3989 df-op 3991 df-uni 4199 df-int 4236 df-br 4400 df-opab 4458 df-tr 4493 df-eprel 4739 df-po 4748 df-so 4749 df-fr 4786 df-we 4788 df-ord 4829 df-on 4830 df-lim 4831 df-suc 4832 df-om 6586 |
This theorem is referenced by: unblem2 7675 unblem3 7676 |
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