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Mirrors > Home > ILE Home > Th. List > iinss | Unicode version |
Description: Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
iinss |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2554 |
. . . 4
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2 | eliin 3653 |
. . . 4
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3 | 1, 2 | ax-mp 7 |
. . 3
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4 | ssel 2933 |
. . . . 5
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5 | 4 | reximi 2410 |
. . . 4
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6 | r19.36av 2455 |
. . . 4
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7 | 5, 6 | syl 14 |
. . 3
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8 | 3, 7 | syl5bi 141 |
. 2
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9 | 8 | ssrdv 2945 |
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-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-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
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-v 2553 df-in 2918 df-ss 2925 df-iin 3651 |
This theorem is referenced by: riinm 3720 reliin 4402 cnviinm 4802 iinerm 6114 |
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