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Mirrors > Home > NFE Home > Th. List > addcid1 | Unicode version |
Description: Cardinal zero is a fixed point for cardinal addition. Theorem X.1.8 of [Rosser] p. 276. (Contributed by SF, 16-Jan-2015.) |
Ref | Expression |
---|---|
addcid1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-0c 4378 |
. . 3
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2 | 1 | addceq2i 4388 |
. 2
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3 | 0ex 4111 |
. . . . . . 7
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4 | ineq2 3452 |
. . . . . . . . . 10
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5 | 4 | eqeq1d 2361 |
. . . . . . . . 9
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6 | uneq2 3413 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 6 | eqeq2d 2364 |
. . . . . . . . 9
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8 | 5, 7 | anbi12d 691 |
. . . . . . . 8
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9 | in0 3577 |
. . . . . . . . 9
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10 | 9 | biantrur 492 |
. . . . . . . 8
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11 | 8, 10 | syl6bbr 254 |
. . . . . . 7
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12 | 3, 11 | rexsn 3769 |
. . . . . 6
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13 | un0 3576 |
. . . . . . 7
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14 | 13 | eqeq2i 2363 |
. . . . . 6
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15 | equcom 1680 |
. . . . . 6
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16 | 12, 14, 15 | 3bitri 262 |
. . . . 5
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17 | 16 | rexbii 2640 |
. . . 4
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18 | eladdc 4399 |
. . . 4
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19 | risset 2662 |
. . . 4
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20 | 17, 18, 19 | 3bitr4i 268 |
. . 3
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21 | 20 | eqriv 2350 |
. 2
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22 | 2, 21 | eqtri 2373 |
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-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4079 ax-sn 4088 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2479 df-ne 2519 df-ral 2620 df-rex 2621 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-nul 3552 df-pw 3725 df-sn 3742 df-pr 3743 df-opk 4059 df-1c 4137 df-pw1 4138 df-ins2k 4188 df-ins3k 4189 df-imak 4190 df-sik 4193 df-ssetk 4194 df-0c 4378 df-addc 4379 |
This theorem is referenced by: addcid2 4408 1cnnc 4409 nncaddccl 4420 ltfinirr 4458 ltfinp1 4463 lefinlteq 4464 lefinrflx 4468 vfin1cltv 4548 nclenn 6250 ncslesuc 6268 nncdiv3 6278 nnc3n3p1 6279 nchoicelem17 6306 |
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