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Mirrors > Home > MPE Home > Th. List > eleesub | Structured version Unicode version |
Description: Membership of a subtraction mapping in a Euclidean space. (Contributed by Scott Fenton, 17-Jul-2013.) |
Ref | Expression |
---|---|
eleesub.1 |
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Ref | Expression |
---|---|
eleesub |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleesub.1 |
. 2
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2 | fveere 23294 |
. . . . . 6
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3 | fveere 23294 |
. . . . . 6
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4 | resubcl 9779 |
. . . . . 6
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5 | 2, 3, 4 | syl2an 477 |
. . . . 5
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6 | 5 | anandirs 827 |
. . . 4
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7 | 6 | ralrimiva 2827 |
. . 3
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8 | eleenn 23289 |
. . . . 5
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9 | mptelee 23288 |
. . . . 5
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10 | 8, 9 | syl 16 |
. . . 4
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11 | 10 | adantr 465 |
. . 3
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12 | 7, 11 | mpbird 232 |
. 2
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13 | 1, 12 | syl5eqel 2544 |
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 1954 ax-ext 2431 ax-sep 4516 ax-nul 4524 ax-pow 4573 ax-pr 4634 ax-un 6477 ax-cnex 9444 ax-resscn 9445 ax-1cn 9446 ax-icn 9447 ax-addcl 9448 ax-addrcl 9449 ax-mulcl 9450 ax-mulrcl 9451 ax-mulcom 9452 ax-addass 9453 ax-mulass 9454 ax-distr 9455 ax-i2m1 9456 ax-1ne0 9457 ax-1rid 9458 ax-rnegex 9459 ax-rrecex 9460 ax-cnre 9461 ax-pre-lttri 9462 ax-pre-lttrn 9463 ax-pre-ltadd 9464 |
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 2265 df-mo 2266 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2602 df-ne 2647 df-nel 2648 df-ral 2801 df-rex 2802 df-reu 2803 df-rab 2805 df-v 3074 df-sbc 3289 df-csb 3391 df-dif 3434 df-un 3436 df-in 3438 df-ss 3445 df-nul 3741 df-if 3895 df-pw 3965 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4195 df-br 4396 df-opab 4454 df-mpt 4455 df-id 4739 df-po 4744 df-so 4745 df-xp 4949 df-rel 4950 df-cnv 4951 df-co 4952 df-dm 4953 df-rn 4954 df-res 4955 df-ima 4956 df-iota 5484 df-fun 5523 df-fn 5524 df-f 5525 df-f1 5526 df-fo 5527 df-f1o 5528 df-fv 5529 df-riota 6156 df-ov 6198 df-oprab 6199 df-mpt2 6200 df-er 7206 df-map 7321 df-en 7416 df-dom 7417 df-sdom 7418 df-pnf 9526 df-mnf 9527 df-ltxr 9529 df-sub 9703 df-neg 9704 df-ee 23284 |
This theorem is referenced by: (None) |
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