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Mirrors > Home > ILE Home > Th. List > en1bg | Unicode version |
Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Jim Kingdon, 13-Apr-2020.) |
Ref | Expression |
---|---|
en1bg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en1 6623 |
. . 3
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2 | id 19 |
. . . . 5
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3 | unieq 3692 |
. . . . . . 7
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4 | vex 2644 |
. . . . . . . 8
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5 | 4 | unisn 3699 |
. . . . . . 7
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6 | 3, 5 | syl6eq 2148 |
. . . . . 6
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7 | 6 | sneqd 3487 |
. . . . 5
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8 | 2, 7 | eqtr4d 2135 |
. . . 4
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9 | 8 | exlimiv 1545 |
. . 3
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10 | 1, 9 | sylbi 120 |
. 2
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11 | uniexg 4299 |
. . . 4
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12 | ensn1g 6621 |
. . . 4
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13 | 11, 12 | syl 14 |
. . 3
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14 | breq1 3878 |
. . 3
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15 | 13, 14 | syl5ibrcom 156 |
. 2
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16 | 10, 15 | impbid2 142 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-reu 2382 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-id 4153 df-suc 4231 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-1o 6243 df-en 6565 |
This theorem is referenced by: en1uniel 6628 |
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