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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 6763 |
. . . 4
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2 | 1 | adantr 276 |
. . 3
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3 | vex 2752 |
. . . . 5
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4 | imaexg 4994 |
. . . . 5
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5 | 3, 4 | ax-mp 5 |
. . . 4
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6 | imassrn 4993 |
. . . . . 6
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7 | simpr 110 |
. . . . . . 7
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8 | f1f 5433 |
. . . . . . 7
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9 | frn 5386 |
. . . . . . 7
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10 | 7, 8, 9 | 3syl 17 |
. . . . . 6
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11 | 6, 10 | sstrid 3178 |
. . . . 5
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12 | ordom 4618 |
. . . . . . . 8
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13 | ordelss 4391 |
. . . . . . . 8
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14 | 12, 13 | mpan 424 |
. . . . . . 7
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15 | 14 | ad2antlr 489 |
. . . . . 6
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16 | simplr 528 |
. . . . . 6
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17 | f1imaeng 6806 |
. . . . . 6
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18 | 7, 15, 16, 17 | syl3anc 1248 |
. . . . 5
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19 | 11, 18 | jca 306 |
. . . 4
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20 | sseq1 3190 |
. . . . . 6
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21 | breq1 4018 |
. . . . . 6
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22 | 20, 21 | anbi12d 473 |
. . . . 5
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23 | 22 | spcegv 2837 |
. . . 4
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24 | 5, 19, 23 | mpsyl 65 |
. . 3
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25 | 2, 24 | exlimddv 1908 |
. 2
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26 | 25 | ralrimiva 2560 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-iinf 4599 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-iord 4378 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-er 6549 df-en 6755 df-dom 6756 |
This theorem is referenced by: (None) |
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