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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 12724 and bj-omex 12725) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-2inf |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2100 |
. . . 4
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2 | bj-om 12720 |
. . . 4
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3 | 1, 2 | mpbii 147 |
. . 3
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4 | bj-indeq 12712 |
. . . . 5
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5 | sseq1 3070 |
. . . . . . 7
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6 | 5 | imbi2d 229 |
. . . . . 6
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7 | 6 | albidv 1763 |
. . . . 5
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8 | 4, 7 | anbi12d 460 |
. . . 4
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9 | 8 | spcegv 2729 |
. . 3
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10 | 3, 9 | mpd 13 |
. 2
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11 | vex 2644 |
. . . . . 6
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12 | bj-om 12720 |
. . . . . 6
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13 | 11, 12 | ax-mp 7 |
. . . . 5
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14 | 13 | biimpri 132 |
. . . 4
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15 | 14 | eximi 1547 |
. . 3
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16 | isset 2647 |
. . 3
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17 | 15, 16 | sylibr 133 |
. 2
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18 | 10, 17 | impbii 125 |
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 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-nul 3994 ax-pr 4069 ax-un 4293 ax-bd0 12592 ax-bdor 12595 ax-bdex 12598 ax-bdeq 12599 ax-bdel 12600 ax-bdsb 12601 ax-bdsep 12663 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-sn 3480 df-pr 3481 df-uni 3684 df-int 3719 df-suc 4231 df-iom 4443 df-bdc 12620 df-bj-ind 12710 |
This theorem is referenced by: bj-omex 12725 |
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