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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-2inf | Unicode version |
Description: Two formulations of the axiom of infinity (see ax-infvn 14544 and bj-omex 14545) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-2inf |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 |
. . . 4
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2 | bj-om 14540 |
. . . 4
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3 | 1, 2 | mpbii 148 |
. . 3
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4 | bj-indeq 14532 |
. . . . 5
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5 | sseq1 3178 |
. . . . . . 7
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6 | 5 | imbi2d 230 |
. . . . . 6
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7 | 6 | albidv 1824 |
. . . . 5
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8 | 4, 7 | anbi12d 473 |
. . . 4
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9 | 8 | spcegv 2825 |
. . 3
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10 | 3, 9 | mpd 13 |
. 2
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11 | vex 2740 |
. . . . . 6
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12 | bj-om 14540 |
. . . . . 6
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13 | 11, 12 | ax-mp 5 |
. . . . 5
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14 | 13 | biimpri 133 |
. . . 4
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15 | 14 | eximi 1600 |
. . 3
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16 | isset 2743 |
. . 3
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17 | 15, 16 | sylibr 134 |
. 2
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18 | 10, 17 | impbii 126 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-nul 4128 ax-pr 4208 ax-un 4432 ax-bd0 14416 ax-bdor 14419 ax-bdex 14422 ax-bdeq 14423 ax-bdel 14424 ax-bdsb 14425 ax-bdsep 14487 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-sn 3598 df-pr 3599 df-uni 3810 df-int 3845 df-suc 4370 df-iom 4589 df-bdc 14444 df-bj-ind 14530 |
This theorem is referenced by: bj-omex 14545 |
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