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Mirrors > Home > MPE Home > Th. List > en1b | Structured version Unicode version |
Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) |
Ref | Expression |
---|---|
en1b |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en1 7481 |
. . 3
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2 | id 22 |
. . . . 5
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3 | unieq 4202 |
. . . . . . 7
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4 | vex 3075 |
. . . . . . . 8
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5 | 4 | unisn 4209 |
. . . . . . 7
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6 | 3, 5 | syl6eq 2509 |
. . . . . 6
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7 | 6 | sneqd 3992 |
. . . . 5
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8 | 2, 7 | eqtr4d 2496 |
. . . 4
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9 | 8 | exlimiv 1689 |
. . 3
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10 | 1, 9 | sylbi 195 |
. 2
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11 | id 22 |
. . 3
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12 | snex 4636 |
. . . . . 6
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13 | 11, 12 | syl6eqel 2548 |
. . . . 5
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14 | uniexg 6482 |
. . . . 5
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15 | 13, 14 | syl 16 |
. . . 4
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16 | ensn1g 7479 |
. . . 4
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17 | 15, 16 | syl 16 |
. . 3
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18 | 11, 17 | eqbrtrd 4415 |
. 2
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19 | 10, 18 | impbii 188 |
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-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1954 ax-ext 2431 ax-sep 4516 ax-nul 4524 ax-pr 4634 ax-un 6477 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2265 df-mo 2266 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2602 df-ne 2647 df-ral 2801 df-rex 2802 df-reu 2803 df-rab 2805 df-v 3074 df-sbc 3289 df-dif 3434 df-un 3436 df-in 3438 df-ss 3445 df-nul 3741 df-if 3895 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4195 df-br 4396 df-opab 4454 df-id 4739 df-suc 4828 df-xp 4949 df-rel 4950 df-cnv 4951 df-co 4952 df-dm 4953 df-rn 4954 df-res 4955 df-ima 4956 df-iota 5484 df-fun 5523 df-fn 5524 df-f 5525 df-f1 5526 df-fo 5527 df-f1o 5528 df-fv 5529 df-1o 7025 df-en 7416 |
This theorem is referenced by: en1uniel 7486 sylow2alem2 16233 sylow2a 16234 frgpcyg 18126 ptcmplem3 19753 cnextfvval 19764 cnextcn 19766 minveclem4a 21044 isppw 22580 xrge0tsmsbi 26394 |
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