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Mirrors > Home > ILE Home > Th. List > ssex | Unicode version |
Description: The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22. This is one way to express the Axiom of Separation ax-sep 4133 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.) |
Ref | Expression |
---|---|
ssex.1 |
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Ref | Expression |
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ssex |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ss 3154 |
. 2
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2 | ssex.1 |
. . . 4
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3 | 2 | inex2 4150 |
. . 3
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4 | eleq1 2250 |
. . 3
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5 | 3, 4 | mpbii 148 |
. 2
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6 | 1, 5 | sylbi 121 |
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-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 ax-sep 4133 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-in 3147 df-ss 3154 |
This theorem is referenced by: ssexi 4153 ssexg 4154 inteximm 4161 exmid1stab 4220 funimaexglem 5311 tfrexlem 6349 elinp 7487 suplocexprlem2b 7727 negfi 11250 ssomct 12460 ssnnctlemct 12461 nninfdc 12468 elcncf 14413 sbthom 15128 |
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