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Mirrors > Home > NFE Home > Th. List > dfss3 | GIF version |
Description: Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.) |
Ref | Expression |
---|---|
dfss3 | ⊢ (A ⊆ B ↔ ∀x ∈ A x ∈ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 3262 | . 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B)) | |
2 | df-ral 2619 | . 2 ⊢ (∀x ∈ A x ∈ B ↔ ∀x(x ∈ A → x ∈ B)) | |
3 | 1, 2 | bitr4i 243 | 1 ⊢ (A ⊆ B ↔ ∀x ∈ A x ∈ B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∀wal 1540 ∈ wcel 1710 ∀wral 2614 ⊆ wss 3257 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 |
This theorem is referenced by: ssrab 3344 eqsn 3867 dfpss4 3888 uni0b 3916 uni0c 3917 ssint 3942 ssiinf 4015 sspwuni 4051 rninxp 5060 fnres 5199 eqfnfv3 5394 funimass3 5404 dff3 5420 ffvresb 5431 |
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