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Mirrors > Home > MPE Home > Th. List > oneqmini | Structured version Unicode version |
Description: A way to show that an ordinal number equals the minimum of a collection of ordinal numbers: it must be in the collection, and it must not be larger than any member of the collection. (Contributed by NM, 14-Nov-2003.) |
Ref | Expression |
---|---|
oneqmini |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssint 4245 |
. . . . . 6
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2 | ssel 3451 |
. . . . . . . . . . . 12
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3 | ssel 3451 |
. . . . . . . . . . . 12
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4 | 2, 3 | anim12d 563 |
. . . . . . . . . . 11
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5 | ontri1 4854 |
. . . . . . . . . . 11
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6 | 4, 5 | syl6 33 |
. . . . . . . . . 10
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7 | 6 | expdimp 437 |
. . . . . . . . 9
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8 | 7 | pm5.74d 247 |
. . . . . . . 8
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9 | con2b 334 |
. . . . . . . 8
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10 | 8, 9 | syl6bb 261 |
. . . . . . 7
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11 | 10 | ralbidv2 2830 |
. . . . . 6
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12 | 1, 11 | syl5bb 257 |
. . . . 5
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13 | 12 | biimprd 223 |
. . . 4
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14 | 13 | expimpd 603 |
. . 3
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15 | intss1 4244 |
. . . . 5
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16 | 15 | a1i 11 |
. . . 4
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17 | 16 | adantrd 468 |
. . 3
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18 | 14, 17 | jcad 533 |
. 2
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19 | eqss 3472 |
. 2
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20 | 18, 19 | syl6ibr 227 |
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 1952 ax-ext 2430 ax-sep 4514 ax-nul 4522 ax-pr 4632 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 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-sbc 3288 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-pss 3445 df-nul 3739 df-if 3893 df-sn 3979 df-pr 3981 df-op 3985 df-uni 4193 df-int 4230 df-br 4394 df-opab 4452 df-tr 4487 df-eprel 4733 df-po 4742 df-so 4743 df-fr 4780 df-we 4782 df-ord 4823 df-on 4824 |
This theorem is referenced by: oneqmin 6519 alephval3 8384 cfsuc 8530 alephval2 8840 |
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