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Mirrors > Home > NFE Home > Th. List > scancan | Unicode version |
Description: Strongly Cantorian implies Cantorian. Observation from [Holmes], p. 134. (Contributed by Scott Fenton, 19-Apr-2021.) |
Ref | Expression |
---|---|
scancan | SCan Can |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 4111 | . . . . . 6 | |
2 | eqid 2353 | . . . . . 6 | |
3 | 1, 2 | fnmpti 5690 | . . . . 5 |
4 | elpw1 4144 | . . . . . . 7 1 | |
5 | euequ1 2292 | . . . . . . . . 9 | |
6 | eqeq1 2359 | . . . . . . . . . . 11 | |
7 | vex 2862 | . . . . . . . . . . . . 13 | |
8 | 7 | sneqb 3876 | . . . . . . . . . . . 12 |
9 | equcom 1680 | . . . . . . . . . . . 12 | |
10 | 8, 9 | bitri 240 | . . . . . . . . . . 11 |
11 | 6, 10 | syl6bb 252 | . . . . . . . . . 10 |
12 | 11 | eubidv 2212 | . . . . . . . . 9 |
13 | 5, 12 | mpbiri 224 | . . . . . . . 8 |
14 | 13 | rexlimivw 2734 | . . . . . . 7 |
15 | 4, 14 | sylbi 187 | . . . . . 6 1 |
16 | df-mpt 5652 | . . . . . . . 8 | |
17 | 16 | cnveqi 4887 | . . . . . . 7 |
18 | cnvopab 5030 | . . . . . . 7 | |
19 | eleq1 2413 | . . . . . . . . . 10 1 1 | |
20 | snelpw1 4146 | . . . . . . . . . 10 1 | |
21 | 19, 20 | syl6rbb 253 | . . . . . . . . 9 1 |
22 | 21 | pm5.32ri 619 | . . . . . . . 8 1 |
23 | 22 | opabbii 4626 | . . . . . . 7 1 |
24 | 17, 18, 23 | 3eqtri 2377 | . . . . . 6 1 |
25 | 15, 24 | fnopab 5207 | . . . . 5 1 |
26 | dff1o4 5294 | . . . . 5 1 1 | |
27 | 3, 25, 26 | mpbir2an 886 | . . . 4 1 |
28 | f1oeng 6032 | . . . 4 1 1 | |
29 | 27, 28 | mpan2 652 | . . 3 1 |
30 | ensymi 6036 | . . 3 1 1 | |
31 | 29, 30 | syl 15 | . 2 1 |
32 | elscan 6330 | . 2 SCan | |
33 | elcan 6329 | . 2 Can 1 | |
34 | 31, 32, 33 | 3imtr4i 257 | 1 SCan Can |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 358 wceq 1642 wcel 1710 weu 2204 wrex 2615 cvv 2859 csn 3737 1 cpw1 4135 copab 4622 class class class wbr 4639 ccnv 4771 wfn 4776 wf1o 4780 cmpt 5651 cen 6028 Can ccan 6323 SCan cscan 6325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-swap 4724 df-co 4726 df-ima 4727 df-id 4767 df-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 df-mpt 5652 df-en 6029 df-can 6324 df-scan 6326 |
This theorem is referenced by: (None) |
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