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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 6707 | . . . 4 | |
2 | 1 | biimpi 119 | . . 3 |
3 | 2 | ad2antrr 480 | . 2 |
4 | isfi 6707 | . . . . 5 | |
5 | 4 | biimpi 119 | . . . 4 |
6 | 5 | ad3antlr 485 | . . 3 |
7 | simplrr 526 | . . . . 5 | |
8 | 7 | ensymd 6729 | . . . . . . . . 9 |
9 | simprl 521 | . . . . . . . . . 10 | |
10 | 9 | ad2antrr 480 | . . . . . . . . 9 |
11 | endomtr 6736 | . . . . . . . . 9 | |
12 | 8, 10, 11 | syl2anc 409 | . . . . . . . 8 |
13 | simprr 522 | . . . . . . . 8 | |
14 | domentr 6737 | . . . . . . . 8 | |
15 | 12, 13, 14 | syl2anc 409 | . . . . . . 7 |
16 | simplrl 525 | . . . . . . . 8 | |
17 | simprl 521 | . . . . . . . 8 | |
18 | nndomo 6810 | . . . . . . . 8 | |
19 | 16, 17, 18 | syl2anc 409 | . . . . . . 7 |
20 | 15, 19 | mpbid 146 | . . . . . 6 |
21 | 13 | ensymd 6729 | . . . . . . . . 9 |
22 | simprr 522 | . . . . . . . . . 10 | |
23 | 22 | ad2antrr 480 | . . . . . . . . 9 |
24 | endomtr 6736 | . . . . . . . . 9 | |
25 | 21, 23, 24 | syl2anc 409 | . . . . . . . 8 |
26 | domentr 6737 | . . . . . . . 8 | |
27 | 25, 7, 26 | syl2anc 409 | . . . . . . 7 |
28 | nndomo 6810 | . . . . . . . 8 | |
29 | 17, 16, 28 | syl2anc 409 | . . . . . . 7 |
30 | 27, 29 | mpbid 146 | . . . . . 6 |
31 | 20, 30 | eqssd 3145 | . . . . 5 |
32 | 7, 31 | breqtrd 3991 | . . . 4 |
33 | entr 6730 | . . . 4 | |
34 | 32, 21, 33 | syl2anc 409 | . . 3 |
35 | 6, 34 | rexlimddv 2579 | . 2 |
36 | 3, 35 | rexlimddv 2579 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2128 wrex 2436 wss 3102 class class class wbr 3966 com 4550 cen 6684 cdom 6685 cfn 6686 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-br 3967 df-opab 4027 df-tr 4064 df-id 4254 df-iord 4327 df-on 4329 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-er 6481 df-en 6687 df-dom 6688 df-fin 6689 |
This theorem is referenced by: (None) |
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