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Mirrors > Home > ILE Home > Th. List > onunsnss | Unicode version |
Description: Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.) |
Ref | Expression |
---|---|
onunsnss |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elirr 4555 |
. . . . 5
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2 | elsni 3625 |
. . . . . . . 8
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3 | 2 | adantl 277 |
. . . . . . 7
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4 | simplr 528 |
. . . . . . 7
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5 | 3, 4 | eqeltrrd 2267 |
. . . . . 6
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6 | 5 | ex 115 |
. . . . 5
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7 | 1, 6 | mtoi 665 |
. . . 4
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8 | snidg 3636 |
. . . . . . . . 9
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9 | elun2 3318 |
. . . . . . . . 9
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10 | 8, 9 | syl 14 |
. . . . . . . 8
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11 | 10 | adantr 276 |
. . . . . . 7
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12 | ontr1 4404 |
. . . . . . . 8
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13 | 12 | adantl 277 |
. . . . . . 7
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14 | 11, 13 | mpan2d 428 |
. . . . . 6
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15 | 14 | imp 124 |
. . . . 5
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16 | elun 3291 |
. . . . 5
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17 | 15, 16 | sylib 122 |
. . . 4
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18 | 7, 17 | ecased 1360 |
. . 3
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19 | 18 | ex 115 |
. 2
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20 | 19 | 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-in1 615 ax-in2 616 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-ext 2171 ax-setind 4551 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-uni 3825 df-tr 4117 df-iord 4381 df-on 4383 |
This theorem is referenced by: (None) |
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