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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 15171 and bj-omex 15172) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-2inf |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2189 |
. . . 4
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2 | bj-om 15167 |
. . . 4
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3 | 1, 2 | mpbii 148 |
. . 3
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4 | bj-indeq 15159 |
. . . . 5
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5 | sseq1 3193 |
. . . . . . 7
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6 | 5 | imbi2d 230 |
. . . . . 6
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7 | 6 | albidv 1835 |
. . . . 5
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8 | 4, 7 | anbi12d 473 |
. . . 4
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9 | 8 | spcegv 2840 |
. . 3
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10 | 3, 9 | mpd 13 |
. 2
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11 | vex 2755 |
. . . . . 6
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12 | bj-om 15167 |
. . . . . 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 1611 |
. . 3
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16 | isset 2758 |
. . 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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-nul 4144 ax-pr 4227 ax-un 4451 ax-bd0 15043 ax-bdor 15046 ax-bdex 15049 ax-bdeq 15050 ax-bdel 15051 ax-bdsb 15052 ax-bdsep 15114 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-sn 3613 df-pr 3614 df-uni 3825 df-int 3860 df-suc 4389 df-iom 4608 df-bdc 15071 df-bj-ind 15157 |
This theorem is referenced by: bj-omex 15172 |
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