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Mirrors > Home > NFE Home > Th. List > nchoicelem5 | Unicode version |
Description: Lemma for nchoice 6309. A cardinal is not a member of the special set of itself raised to two. Theorem 6.5 of [Specker] p. 973. (Contributed by SF, 13-Mar-2015.) |
Ref | Expression |
---|---|
nchoicelem5 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ce2lt 6221 |
. . . 4
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2 | 2nc 6169 |
. . . . . . 7
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3 | 2 | jctl 525 |
. . . . . 6
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4 | 2nnc 6168 |
. . . . . . . . 9
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5 | ce0nn 6181 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 4, 5 | ax-mp 5 |
. . . . . . . 8
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7 | ce0nulnc 6185 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 2, 7 | ax-mp 5 |
. . . . . . . 8
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9 | 6, 8 | mpbi 199 |
. . . . . . 7
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10 | 9 | jctl 525 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | cecl 6187 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 3, 10, 11 | syl2an 463 |
. . . . 5
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13 | ltlenlec 6208 |
. . . . 5
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14 | 12, 13 | syldan 456 |
. . . 4
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15 | 1, 14 | mpbid 201 |
. . 3
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16 | 15 | simprd 449 |
. 2
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17 | nchoicelem4 6293 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | 12, 17 | sylan 457 |
. 2
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19 | 16, 18 | mtand 640 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 4079 ax-xp 4080 ax-cnv 4081 ax-1c 4082 ax-sset 4083 ax-si 4084 ax-ins2 4085 ax-ins3 4086 ax-typlower 4087 ax-sn 4088 |
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 2479 df-ne 2519 df-ral 2620 df-rex 2621 df-reu 2622 df-rmo 2623 df-rab 2624 df-v 2862 df-sbc 3048 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-symdif 3217 df-ss 3260 df-pss 3262 df-nul 3552 df-if 3664 df-pw 3725 df-sn 3742 df-pr 3743 df-uni 3893 df-int 3928 df-opk 4059 df-1c 4137 df-pw1 4138 df-uni1 4139 df-xpk 4186 df-cnvk 4187 df-ins2k 4188 df-ins3k 4189 df-imak 4190 df-cok 4191 df-p6 4192 df-sik 4193 df-ssetk 4194 df-imagek 4195 df-idk 4196 df-iota 4340 df-0c 4378 df-addc 4379 df-nnc 4380 df-fin 4381 df-lefin 4441 df-ltfin 4442 df-ncfin 4443 df-tfin 4444 df-evenfin 4445 df-oddfin 4446 df-sfin 4447 df-spfin 4448 df-phi 4566 df-op 4567 df-proj1 4568 df-proj2 4569 df-opab 4624 df-br 4641 df-1st 4724 df-swap 4725 df-sset 4726 df-co 4727 df-ima 4728 df-si 4729 df-id 4768 df-xp 4785 df-cnv 4786 df-rn 4787 df-dm 4788 df-res 4789 df-fun 4790 df-fn 4791 df-f 4792 df-f1 4793 df-fo 4794 df-f1o 4795 df-fv 4796 df-2nd 4798 df-ov 5527 df-oprab 5529 df-mpt 5653 df-mpt2 5655 df-txp 5737 df-fix 5741 df-compose 5749 df-ins2 5751 df-ins3 5753 df-image 5755 df-ins4 5757 df-si3 5759 df-funs 5761 df-fns 5763 df-pw1fn 5767 df-fullfun 5769 df-clos1 5874 df-trans 5900 df-sym 5909 df-er 5910 df-ec 5948 df-qs 5952 df-map 6002 df-en 6030 df-ncs 6099 df-lec 6100 df-ltc 6101 df-nc 6102 df-2c 6105 df-ce 6107 df-spac 6275 |
This theorem is referenced by: nchoicelem7 6296 nchoicelem14 6303 |
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