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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 9401 and bj-omex 9402) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-2inf |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2037 |
. . . 4
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2 | bj-om 9396 |
. . . 4
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3 | 1, 2 | mpbii 136 |
. . 3
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4 | bj-indeq 9388 |
. . . . 5
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5 | sseq1 2960 |
. . . . . . 7
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6 | 5 | imbi2d 219 |
. . . . . 6
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7 | 6 | albidv 1702 |
. . . . 5
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8 | 4, 7 | anbi12d 442 |
. . . 4
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9 | 8 | spcegv 2635 |
. . 3
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10 | 3, 9 | mpd 13 |
. 2
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11 | vex 2554 |
. . . . . 6
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12 | bj-om 9396 |
. . . . . 6
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13 | 11, 12 | ax-mp 7 |
. . . . 5
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14 | 13 | biimpri 124 |
. . . 4
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15 | 14 | eximi 1488 |
. . 3
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16 | isset 2555 |
. . 3
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17 | 15, 16 | sylibr 137 |
. 2
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18 | 10, 17 | impbii 117 |
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 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-nul 3874 ax-pr 3935 ax-un 4136 ax-bd0 9268 ax-bdor 9271 ax-bdex 9274 ax-bdeq 9275 ax-bdel 9276 ax-bdsb 9277 ax-bdsep 9339 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-sn 3373 df-pr 3374 df-uni 3572 df-int 3607 df-suc 4074 df-iom 4257 df-bdc 9296 df-bj-ind 9386 |
This theorem is referenced by: bj-omex 9402 |
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