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Mirrors > Home > MPE Home > Th. List > sdomdomtr | Structured version Visualization version Unicode version |
Description: Transitivity of strict dominance and dominance. Theorem 22(iii) of [Suppes] p. 97. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
sdomdomtr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdomdom 7594 |
. . 3
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2 | domtr 7619 |
. . 3
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3 | 1, 2 | sylan 474 |
. 2
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4 | simpl 459 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | simpr 463 |
. . . . . 6
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6 | ensym 7615 |
. . . . . 6
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7 | domentr 7625 |
. . . . . 6
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8 | 5, 6, 7 | syl2an 480 |
. . . . 5
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9 | domnsym 7695 |
. . . . 5
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10 | 8, 9 | syl 17 |
. . . 4
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11 | 10 | ex 436 |
. . 3
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12 | 4, 11 | mt2d 121 |
. 2
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13 | brsdom 7589 |
. 2
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14 | 3, 12, 13 | sylanbrc 669 |
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 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-ral 2741 df-rex 2742 df-rab 2745 df-v 3046 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-op 3974 df-uni 4198 df-br 4402 df-opab 4461 df-id 4748 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-er 7360 df-en 7567 df-dom 7568 df-sdom 7569 |
This theorem is referenced by: sdomentr 7703 sucdom 7766 infsdomnn 7829 fodomfib 7848 marypha1lem 7944 r1sdom 8242 infxpenlem 8441 infunsdom1 8640 fin56 8820 fodomb 8951 pwcfsdom 9005 cfpwsdom 9006 canthp1lem2 9075 gchpwdom 9092 gchhar 9101 gchina 9121 tsksdom 9178 tskpr 9192 tskcard 9203 gruina 9240 |
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