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Mirrors > Home > MPE Home > Th. List > tcel | Structured version Unicode version |
Description: The transitive closure function converts the element relation to the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.) |
Ref | Expression |
---|---|
tc2.1 |
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Ref | Expression |
---|---|
tcel |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tcvalg 8070 |
. 2
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2 | ssel 3459 |
. . . . . . . 8
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3 | trss 4503 |
. . . . . . . . 9
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4 | 3 | com12 31 |
. . . . . . . 8
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5 | 2, 4 | syl6com 35 |
. . . . . . 7
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6 | 5 | impd 431 |
. . . . . 6
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7 | simpr 461 |
. . . . . . 7
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8 | 7 | a1i 11 |
. . . . . 6
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9 | 6, 8 | jcad 533 |
. . . . 5
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10 | 9 | ss2abdv 3534 |
. . . 4
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11 | intss 4258 |
. . . 4
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12 | 10, 11 | syl 16 |
. . 3
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13 | tc2.1 |
. . . 4
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14 | tcvalg 8070 |
. . . 4
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15 | 13, 14 | ax-mp 5 |
. . 3
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16 | 12, 15 | syl6sseqr 3512 |
. 2
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17 | 1, 16 | eqsstrd 3499 |
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-rep 4512 ax-sep 4522 ax-nul 4530 ax-pow 4579 ax-pr 4640 ax-un 6483 ax-inf2 7959 |
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 2650 df-ral 2804 df-rex 2805 df-reu 2806 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3397 df-dif 3440 df-un 3442 df-in 3444 df-ss 3451 df-pss 3453 df-nul 3747 df-if 3901 df-pw 3971 df-sn 3987 df-pr 3989 df-tp 3991 df-op 3993 df-uni 4201 df-int 4238 df-iun 4282 df-br 4402 df-opab 4460 df-mpt 4461 df-tr 4495 df-eprel 4741 df-id 4745 df-po 4750 df-so 4751 df-fr 4788 df-we 4790 df-ord 4831 df-on 4832 df-lim 4833 df-suc 4834 df-xp 4955 df-rel 4956 df-cnv 4957 df-co 4958 df-dm 4959 df-rn 4960 df-res 4961 df-ima 4962 df-iota 5490 df-fun 5529 df-fn 5530 df-f 5531 df-f1 5532 df-fo 5533 df-f1o 5534 df-fv 5535 df-om 6588 df-recs 6943 df-rdg 6977 df-tc 8069 |
This theorem is referenced by: tcrank 8203 hsmexlem4 8710 |
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