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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pclssN | Structured version Unicode version |
Description: Ordering is preserved by subspace closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pclss.a |
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pclss.c |
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Ref | Expression |
---|---|
pclssN |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstr2 3458 |
. . . . . 6
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2 | 1 | 3ad2ant2 1010 |
. . . . 5
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3 | 2 | adantr 465 |
. . . 4
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4 | 3 | ss2rabdv 3528 |
. . 3
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5 | intss 4244 |
. . 3
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6 | 4, 5 | syl 16 |
. 2
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7 | simp1 988 |
. . 3
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8 | sstr 3459 |
. . . 4
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9 | 8 | 3adant1 1006 |
. . 3
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10 | pclss.a |
. . . 4
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11 | eqid 2451 |
. . . 4
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12 | pclss.c |
. . . 4
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13 | 10, 11, 12 | pclvalN 33837 |
. . 3
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14 | 7, 9, 13 | syl2anc 661 |
. 2
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15 | 10, 11, 12 | pclvalN 33837 |
. . 3
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16 | 15 | 3adant2 1007 |
. 2
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17 | 6, 14, 16 | 3sstr4d 3494 |
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 1952 ax-ext 2430 ax-rep 4498 ax-sep 4508 ax-nul 4516 ax-pow 4565 ax-pr 4626 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2599 df-ne 2644 df-ral 2798 df-rex 2799 df-reu 2800 df-rab 2802 df-v 3067 df-sbc 3282 df-csb 3384 df-dif 3426 df-un 3428 df-in 3430 df-ss 3437 df-nul 3733 df-if 3887 df-pw 3957 df-sn 3973 df-pr 3975 df-op 3979 df-uni 4187 df-int 4224 df-iun 4268 df-br 4388 df-opab 4446 df-mpt 4447 df-id 4731 df-xp 4941 df-rel 4942 df-cnv 4943 df-co 4944 df-dm 4945 df-rn 4946 df-res 4947 df-ima 4948 df-iota 5476 df-fun 5515 df-fn 5516 df-f 5517 df-f1 5518 df-fo 5519 df-f1o 5520 df-fv 5521 df-ov 6190 df-psubsp 33450 df-pclN 33835 |
This theorem is referenced by: pclbtwnN 33844 pclunN 33845 pclfinN 33847 pclss2polN 33868 pclfinclN 33897 |
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