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Mirrors > Home > MPE Home > Th. List > pmtrdifellem4 | Structured version Visualization version Unicode version |
Description: Lemma 4 for pmtrdifel 17176. (Contributed by AV, 28-Jan-2019.) |
Ref | Expression |
---|---|
pmtrdifel.t |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
pmtrdifel.r |
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pmtrdifel.0 |
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Ref | Expression |
---|---|
pmtrdifellem4 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmtrdifel.t |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | pmtrdifel.r |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | pmtrdifel.0 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | 1, 2, 3 | pmtrdifellem1 17172 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | eqid 2462 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | eqid 2462 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 5, 2, 6 | pmtrffv 17155 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | 4, 7 | sylan 478 |
. 2
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9 | eqid 2462 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | eqid 2462 |
. . . . . . . 8
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11 | 1, 9, 10 | symgtrf 17165 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 11 | sseli 3440 |
. . . . . 6
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13 | 9, 10 | symgbasf 17080 |
. . . . . 6
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14 | ffn 5755 |
. . . . . . 7
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15 | fndifnfp 6122 |
. . . . . . 7
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16 | ssrab2 3526 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | ssel2 3439 |
. . . . . . . . . . 11
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18 | eldif 3426 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | elsncg 4003 |
. . . . . . . . . . . . . . 15
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 19 | notbid 300 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | eqid 2462 |
. . . . . . . . . . . . . . 15
![]() ![]() ![]() ![]() | |
22 | 21 | pm2.24i 138 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 20, 22 | syl6bi 236 |
. . . . . . . . . . . . 13
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24 | 23 | imp 435 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 18, 24 | sylbi 200 |
. . . . . . . . . . 11
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26 | 17, 25 | syl 17 |
. . . . . . . . . 10
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27 | 16, 26 | mpan 681 |
. . . . . . . . 9
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28 | 27 | con2i 125 |
. . . . . . . 8
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29 | eleq2 2529 |
. . . . . . . . 9
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30 | 29 | notbid 300 |
. . . . . . . 8
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31 | 28, 30 | syl5ibr 229 |
. . . . . . 7
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32 | 14, 15, 31 | 3syl 18 |
. . . . . 6
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33 | 12, 13, 32 | 3syl 18 |
. . . . 5
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34 | 33 | imp 435 |
. . . 4
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35 | 1, 2, 3 | pmtrdifellem2 17173 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | 35 | eleq2d 2525 |
. . . . 5
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37 | 36 | adantr 471 |
. . . 4
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38 | 34, 37 | mtbird 307 |
. . 3
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39 | 38 | iffalsed 3904 |
. 2
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40 | 8, 39 | eqtrd 2496 |
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-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-rep 4531 ax-sep 4541 ax-nul 4550 ax-pow 4598 ax-pr 4656 ax-un 6615 ax-cnex 9626 ax-resscn 9627 ax-1cn 9628 ax-icn 9629 ax-addcl 9630 ax-addrcl 9631 ax-mulcl 9632 ax-mulrcl 9633 ax-mulcom 9634 ax-addass 9635 ax-mulass 9636 ax-distr 9637 ax-i2m1 9638 ax-1ne0 9639 ax-1rid 9640 ax-rnegex 9641 ax-rrecex 9642 ax-cnre 9643 ax-pre-lttri 9644 ax-pre-lttrn 9645 ax-pre-ltadd 9646 ax-pre-mulgt0 9647 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-nel 2636 df-ral 2754 df-rex 2755 df-reu 2756 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-pss 3432 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-tp 3985 df-op 3987 df-uni 4213 df-int 4249 df-iun 4294 df-br 4419 df-opab 4478 df-mpt 4479 df-tr 4514 df-eprel 4767 df-id 4771 df-po 4777 df-so 4778 df-fr 4815 df-we 4817 df-xp 4862 df-rel 4863 df-cnv 4864 df-co 4865 df-dm 4866 df-rn 4867 df-res 4868 df-ima 4869 df-pred 5403 df-ord 5449 df-on 5450 df-lim 5451 df-suc 5452 df-iota 5569 df-fun 5607 df-fn 5608 df-f 5609 df-f1 5610 df-fo 5611 df-f1o 5612 df-fv 5613 df-riota 6282 df-ov 6323 df-oprab 6324 df-mpt2 6325 df-om 6725 df-1st 6825 df-2nd 6826 df-wrecs 7059 df-recs 7121 df-rdg 7159 df-1o 7213 df-2o 7214 df-oadd 7217 df-er 7394 df-map 7505 df-en 7601 df-dom 7602 df-sdom 7603 df-fin 7604 df-pnf 9708 df-mnf 9709 df-xr 9710 df-ltxr 9711 df-le 9712 df-sub 9893 df-neg 9894 df-nn 10643 df-2 10701 df-3 10702 df-4 10703 df-5 10704 df-6 10705 df-7 10706 df-8 10707 df-9 10708 df-n0 10904 df-z 10972 df-uz 11194 df-fz 11820 df-struct 15178 df-ndx 15179 df-slot 15180 df-base 15181 df-plusg 15258 df-tset 15264 df-symg 17074 df-pmtr 17138 |
This theorem is referenced by: pmtrdifwrdel2lem1 17180 |
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