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Mirrors > Home > MPE Home > Th. List > ptbasin2 | Structured version Unicode version |
Description: The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 19-Mar-2015.) |
Ref | Expression |
---|---|
ptbas.1 |
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Ref | Expression |
---|---|
ptbasin2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ptbas.1 |
. . . 4
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2 | 1 | ptbasin 19275 |
. . 3
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3 | 2 | ralrimivva 2907 |
. 2
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4 | 1 | ptuni2 19274 |
. . . . 5
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5 | ixpexg 7390 |
. . . . . 6
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6 | fvex 5802 |
. . . . . . . 8
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7 | 6 | uniex 6479 |
. . . . . . 7
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8 | 7 | a1i 11 |
. . . . . 6
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9 | 5, 8 | mprg 2896 |
. . . . 5
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10 | 4, 9 | syl6eqelr 2548 |
. . . 4
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11 | uniexb 6489 |
. . . 4
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12 | 10, 11 | sylibr 212 |
. . 3
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13 | inficl 7779 |
. . 3
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14 | 12, 13 | syl 16 |
. 2
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15 | 3, 14 | mpbid 210 |
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 4504 ax-sep 4514 ax-nul 4522 ax-pow 4571 ax-pr 4632 ax-un 6475 |
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 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-ral 2800 df-rex 2801 df-reu 2802 df-rab 2804 df-v 3073 df-sbc 3288 df-csb 3390 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-pss 3445 df-nul 3739 df-if 3893 df-pw 3963 df-sn 3979 df-pr 3981 df-tp 3983 df-op 3985 df-uni 4193 df-int 4230 df-iun 4274 df-br 4394 df-opab 4452 df-mpt 4453 df-tr 4487 df-eprel 4733 df-id 4737 df-po 4742 df-so 4743 df-fr 4780 df-we 4782 df-ord 4823 df-on 4824 df-lim 4825 df-suc 4826 df-xp 4947 df-rel 4948 df-cnv 4949 df-co 4950 df-dm 4951 df-rn 4952 df-res 4953 df-ima 4954 df-iota 5482 df-fun 5521 df-fn 5522 df-f 5523 df-f1 5524 df-fo 5525 df-f1o 5526 df-fv 5527 df-ov 6196 df-oprab 6197 df-mpt2 6198 df-om 6580 df-recs 6935 df-rdg 6969 df-1o 7023 df-oadd 7027 df-er 7204 df-ixp 7367 df-en 7414 df-fin 7417 df-fi 7765 df-top 18628 |
This theorem is referenced by: ptbas 19277 ptbasfi 19279 |
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