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Mirrors > Home > MPE Home > Th. List > domsdomtr | Structured version Unicode version |
Description: Transitivity of dominance and strict dominance. Theorem 22(ii) of [Suppes] p. 97. (Contributed by NM, 10-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
domsdomtr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdomdom 7440 |
. . 3
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2 | domtr 7465 |
. . 3
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3 | 1, 2 | sylan2 474 |
. 2
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4 | simpr 461 |
. . 3
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5 | ensym 7461 |
. . . . . 6
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6 | simpl 457 |
. . . . . 6
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7 | endomtr 7470 |
. . . . . 6
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8 | 5, 6, 7 | syl2anr 478 |
. . . . 5
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9 | domnsym 7540 |
. . . . 5
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10 | 8, 9 | syl 16 |
. . . 4
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11 | 10 | ex 434 |
. . 3
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12 | 4, 11 | mt2d 117 |
. 2
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13 | brsdom 7435 |
. 2
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14 | 3, 12, 13 | sylanbrc 664 |
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-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-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-rab 2804 df-v 3073 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-nul 3739 df-if 3893 df-pw 3963 df-sn 3979 df-pr 3981 df-op 3985 df-uni 4193 df-br 4394 df-opab 4452 df-id 4737 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-fun 5521 df-fn 5522 df-f 5523 df-f1 5524 df-fo 5525 df-f1o 5526 df-er 7204 df-en 7414 df-dom 7415 df-sdom 7416 |
This theorem is referenced by: ensdomtr 7550 sdomtr 7552 2pwuninel 7569 card2on 7873 tskwe 8224 harval2 8271 prdom2 8277 infxpenlem 8284 alephsucdom 8353 pwsdompw 8477 infunsdom1 8486 fin34 8663 ondomon 8831 cardmin 8832 konigthlem 8836 gchpwdom 8941 gchina 8970 inar1 9046 tskord 9051 tskuni 9054 tskurn 9060 csdfil 19592 |
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