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Mirrors > Home > ILE Home > Th. List > enpr2d | Unicode version |
Description: A pair with distinct elements is equinumerous to ordinal two. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
Ref | Expression |
---|---|
enpr2d.1 |
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enpr2d.2 |
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enpr2d.3 |
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Ref | Expression |
---|---|
enpr2d |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enpr2d.1 |
. . . . 5
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2 | ensn1g 6791 |
. . . . 5
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3 | 1, 2 | syl 14 |
. . . 4
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4 | enpr2d.2 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 1on 6418 |
. . . . 5
![]() ![]() ![]() ![]() | |
6 | en2sn 6807 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 4, 5, 6 | sylancl 413 |
. . . 4
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8 | enpr2d.3 |
. . . . . 6
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9 | 8 | neqned 2354 |
. . . . 5
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10 | disjsn2 3654 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 9, 10 | syl 14 |
. . . 4
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12 | 5 | onirri 4539 |
. . . . . 6
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13 | 12 | a1i 9 |
. . . . 5
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14 | disjsn 3653 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 13, 14 | sylibr 134 |
. . . 4
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16 | unen 6810 |
. . . 4
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17 | 3, 7, 11, 15, 16 | syl22anc 1239 |
. . 3
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18 | df-pr 3598 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | df-suc 4368 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 17, 18, 19 | 3brtr4g 4034 |
. 2
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21 | df-2o 6412 |
. 2
![]() ![]() ![]() ![]() ![]() | |
22 | 20, 21 | breqtrrdi 4042 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-tr 4099 df-id 4290 df-iord 4363 df-on 4365 df-suc 4368 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-1o 6411 df-2o 6412 df-er 6529 df-en 6735 |
This theorem is referenced by: (None) |
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