Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > fisbth | Unicode version |
Description: Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim Kingdon, 12-Sep-2021.) |
Ref | Expression |
---|---|
fisbth |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6648 | . . . 4 | |
2 | 1 | biimpi 119 | . . 3 |
3 | 2 | ad2antrr 479 | . 2 |
4 | isfi 6648 | . . . . 5 | |
5 | 4 | biimpi 119 | . . . 4 |
6 | 5 | ad3antlr 484 | . . 3 |
7 | simplrr 525 | . . . . 5 | |
8 | 7 | ensymd 6670 | . . . . . . . . 9 |
9 | simprl 520 | . . . . . . . . . 10 | |
10 | 9 | ad2antrr 479 | . . . . . . . . 9 |
11 | endomtr 6677 | . . . . . . . . 9 | |
12 | 8, 10, 11 | syl2anc 408 | . . . . . . . 8 |
13 | simprr 521 | . . . . . . . 8 | |
14 | domentr 6678 | . . . . . . . 8 | |
15 | 12, 13, 14 | syl2anc 408 | . . . . . . 7 |
16 | simplrl 524 | . . . . . . . 8 | |
17 | simprl 520 | . . . . . . . 8 | |
18 | nndomo 6751 | . . . . . . . 8 | |
19 | 16, 17, 18 | syl2anc 408 | . . . . . . 7 |
20 | 15, 19 | mpbid 146 | . . . . . 6 |
21 | 13 | ensymd 6670 | . . . . . . . . 9 |
22 | simprr 521 | . . . . . . . . . 10 | |
23 | 22 | ad2antrr 479 | . . . . . . . . 9 |
24 | endomtr 6677 | . . . . . . . . 9 | |
25 | 21, 23, 24 | syl2anc 408 | . . . . . . . 8 |
26 | domentr 6678 | . . . . . . . 8 | |
27 | 25, 7, 26 | syl2anc 408 | . . . . . . 7 |
28 | nndomo 6751 | . . . . . . . 8 | |
29 | 17, 16, 28 | syl2anc 408 | . . . . . . 7 |
30 | 27, 29 | mpbid 146 | . . . . . 6 |
31 | 20, 30 | eqssd 3109 | . . . . 5 |
32 | 7, 31 | breqtrd 3949 | . . . 4 |
33 | entr 6671 | . . . 4 | |
34 | 32, 21, 33 | syl2anc 408 | . . 3 |
35 | 6, 34 | rexlimddv 2552 | . 2 |
36 | 3, 35 | rexlimddv 2552 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 1480 wrex 2415 wss 3066 class class class wbr 3924 com 4499 cen 6625 cdom 6626 cfn 6627 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-er 6422 df-en 6628 df-dom 6629 df-fin 6630 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |