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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 | |
enpr2d.2 | |
enpr2d.3 |
Ref | Expression |
---|---|
enpr2d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enpr2d.1 | . . . . 5 | |
2 | ensn1g 6699 | . . . . 5 | |
3 | 1, 2 | syl 14 | . . . 4 |
4 | enpr2d.2 | . . . . 5 | |
5 | 1on 6328 | . . . . 5 | |
6 | en2sn 6715 | . . . . 5 | |
7 | 4, 5, 6 | sylancl 410 | . . . 4 |
8 | enpr2d.3 | . . . . . 6 | |
9 | 8 | neqned 2316 | . . . . 5 |
10 | disjsn2 3594 | . . . . 5 | |
11 | 9, 10 | syl 14 | . . . 4 |
12 | 5 | onirri 4466 | . . . . . 6 |
13 | 12 | a1i 9 | . . . . 5 |
14 | disjsn 3593 | . . . . 5 | |
15 | 13, 14 | sylibr 133 | . . . 4 |
16 | unen 6718 | . . . 4 | |
17 | 3, 7, 11, 15, 16 | syl22anc 1218 | . . 3 |
18 | df-pr 3539 | . . 3 | |
19 | df-suc 4301 | . . 3 | |
20 | 17, 18, 19 | 3brtr4g 3970 | . 2 |
21 | df-2o 6322 | . 2 | |
22 | 20, 21 | breqtrrdi 3978 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wceq 1332 wcel 1481 wne 2309 cun 3074 cin 3075 c0 3368 csn 3532 cpr 3533 class class class wbr 3937 con0 4293 csuc 4295 c1o 6314 c2o 6315 cen 6640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-1o 6321 df-2o 6322 df-er 6437 df-en 6643 |
This theorem is referenced by: (None) |
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