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Mirrors > Home > MPE Home > Th. List > difin0ss | Structured version Visualization version Unicode version |
Description: Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.) |
Ref | Expression |
---|---|
difin0ss |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eq0 3747 |
. 2
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2 | iman 426 |
. . . . . 6
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3 | elin 3617 |
. . . . . . . 8
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4 | eldif 3414 |
. . . . . . . . 9
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5 | 4 | anbi1i 701 |
. . . . . . . 8
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6 | 3, 5 | bitri 253 |
. . . . . . 7
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7 | ancom 452 |
. . . . . . 7
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8 | annim 427 |
. . . . . . . 8
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9 | 8 | anbi2i 700 |
. . . . . . 7
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10 | 6, 7, 9 | 3bitr2i 277 |
. . . . . 6
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11 | 2, 10 | xchbinxr 313 |
. . . . 5
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12 | ax-2 7 |
. . . . 5
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13 | 11, 12 | sylbir 217 |
. . . 4
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14 | 13 | al2imi 1687 |
. . 3
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15 | dfss2 3421 |
. . 3
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16 | dfss2 3421 |
. . 3
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17 | 14, 15, 16 | 3imtr4g 274 |
. 2
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18 | 1, 17 | sylbi 199 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-v 3047 df-dif 3407 df-in 3411 df-ss 3418 df-nul 3732 |
This theorem is referenced by: tz7.7 5449 tfi 6680 lebnumlem3 21991 lebnumlem3OLD 21994 |
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