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Mirrors > Home > ILE Home > Th. List > 2basgeng | Unicode version |
Description: Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Jim Kingdon, 5-Mar-2023.) |
Ref | Expression |
---|---|
2basgeng |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgvalex 12729 |
. . . . 5
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2 | 1 | 3ad2ant1 1019 |
. . . 4
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3 | simp3 1000 |
. . . 4
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4 | 2, 3 | ssexd 4155 |
. . 3
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5 | simp2 999 |
. . 3
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6 | tgss 13803 |
. . 3
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7 | 4, 5, 6 | syl2anc 411 |
. 2
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8 | simp1 998 |
. . . 4
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9 | tgss3 13818 |
. . . 4
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10 | 4, 8, 9 | syl2anc 411 |
. . 3
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11 | 3, 10 | mpbird 167 |
. 2
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12 | 7, 11 | eqssd 3184 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-sbc 2975 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-topgen 12726 |
This theorem is referenced by: txbasval 14007 tgioo 14286 tgqioo 14287 |
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