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Mirrors > Home > MPE Home > Th. List > gchi | Structured version Unicode version |
Description: The only GCH-sets which have other sets between it and its power set are finite sets. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
gchi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relsdom 7430 |
. . . . . . 7
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2 | 1 | brrelexi 4990 |
. . . . . 6
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3 | 2 | adantl 466 |
. . . . 5
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4 | breq2 4407 |
. . . . . . 7
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5 | breq1 4406 |
. . . . . . 7
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6 | 4, 5 | anbi12d 710 |
. . . . . 6
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7 | 6 | spcegv 3164 |
. . . . 5
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8 | 3, 7 | mpcom 36 |
. . . 4
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9 | df-ex 1588 |
. . . 4
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10 | 8, 9 | sylib 196 |
. . 3
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11 | elgch 8903 |
. . . . . 6
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12 | 11 | ibi 241 |
. . . . 5
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13 | 12 | orcomd 388 |
. . . 4
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14 | 13 | ord 377 |
. . 3
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15 | 10, 14 | syl5 32 |
. 2
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16 | 15 | 3impib 1186 |
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-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4524 ax-nul 4532 ax-pr 4642 |
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-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-ral 2804 df-rex 2805 df-rab 2808 df-v 3080 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-op 3995 df-br 4404 df-opab 4462 df-xp 4957 df-rel 4958 df-dom 7425 df-sdom 7426 df-gch 8902 |
This theorem is referenced by: gchen1 8906 gchen2 8907 gchpwdom 8951 gchaleph 8952 |
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