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Mirrors > Home > MPE Home > Th. List > onminsb | Structured version Visualization version Unicode version |
Description: If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses implicit substitution. Theorem Schema 62 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.) |
Ref | Expression |
---|---|
onminsb.1 |
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onminsb.2 |
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Ref | Expression |
---|---|
onminsb |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabn0 3754 |
. . 3
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2 | ssrab2 3516 |
. . . 4
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3 | onint 6627 |
. . . 4
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4 | 2, 3 | mpan 677 |
. . 3
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5 | 1, 4 | sylbir 217 |
. 2
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6 | nfrab1 2973 |
. . . . 5
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7 | 6 | nfint 4247 |
. . . 4
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8 | nfcv 2594 |
. . . 4
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9 | onminsb.1 |
. . . 4
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10 | onminsb.2 |
. . . 4
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11 | 7, 8, 9, 10 | elrabf 3196 |
. . 3
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12 | 11 | simprbi 466 |
. 2
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13 | 5, 12 | syl 17 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 ax-nul 4537 ax-pr 4642 ax-un 6588 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-rab 2748 df-v 3049 df-sbc 3270 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-pss 3422 df-nul 3734 df-if 3884 df-sn 3971 df-pr 3973 df-tp 3975 df-op 3977 df-uni 4202 df-int 4238 df-br 4406 df-opab 4465 df-tr 4501 df-eprel 4748 df-po 4758 df-so 4759 df-fr 4796 df-we 4798 df-ord 5429 df-on 5430 |
This theorem is referenced by: oawordeulem 7260 rankidb 8276 cardmin2 8437 cardaleph 8525 cardmin 8994 |
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