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Mirrors > Home > ILE Home > Th. List > smoiun | Unicode version |
Description: The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.) |
Ref | Expression |
---|---|
smoiun |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliun 3905 |
. . 3
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2 | smofvon 6325 |
. . . . 5
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3 | smoel 6326 |
. . . . . 6
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4 | 3 | 3expia 1207 |
. . . . 5
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5 | ontr1 4407 |
. . . . . 6
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6 | 5 | expcomd 1452 |
. . . . 5
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7 | 2, 4, 6 | sylsyld 58 |
. . . 4
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8 | 7 | rexlimdv 2606 |
. . 3
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9 | 1, 8 | biimtrid 152 |
. 2
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10 | 9 | ssrdv 3176 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-smo 6312 |
This theorem is referenced by: (None) |
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