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Mirrors > Home > ILE Home > Th. List > isinfinf | Unicode version |
Description: An infinite set contains subsets of arbitrarily large finite cardinality. (Contributed by Jim Kingdon, 15-Jun-2022.) |
Ref | Expression |
---|---|
isinfinf |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdomi 6520 |
. . . 4
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2 | 1 | adantr 271 |
. . 3
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3 | vex 2623 |
. . . . 5
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4 | imaexg 4799 |
. . . . 5
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5 | 3, 4 | ax-mp 7 |
. . . 4
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6 | imassrn 4798 |
. . . . . 6
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7 | simpr 109 |
. . . . . . 7
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8 | f1f 5229 |
. . . . . . 7
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9 | frn 5182 |
. . . . . . 7
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10 | 7, 8, 9 | 3syl 17 |
. . . . . 6
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11 | 6, 10 | syl5ss 3037 |
. . . . 5
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12 | ordom 4434 |
. . . . . . . 8
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13 | ordelss 4215 |
. . . . . . . 8
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14 | 12, 13 | mpan 416 |
. . . . . . 7
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15 | 14 | ad2antlr 474 |
. . . . . 6
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16 | simplr 498 |
. . . . . 6
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17 | f1imaeng 6563 |
. . . . . 6
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18 | 7, 15, 16, 17 | syl3anc 1175 |
. . . . 5
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19 | 11, 18 | jca 301 |
. . . 4
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20 | sseq1 3048 |
. . . . . 6
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21 | breq1 3854 |
. . . . . 6
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22 | 20, 21 | anbi12d 458 |
. . . . 5
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23 | 22 | spcegv 2708 |
. . . 4
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24 | 5, 19, 23 | mpsyl 65 |
. . 3
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25 | 2, 24 | exlimddv 1827 |
. 2
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26 | 25 | ralrimiva 2447 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-iinf 4416 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-iord 4202 df-suc 4207 df-iom 4419 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-res 4463 df-ima 4464 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-er 6306 df-en 6512 df-dom 6513 |
This theorem is referenced by: (None) |
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