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Mirrors > Home > NFE Home > Th. List > dfss2 | GIF version |
Description: Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.) |
Ref | Expression |
---|---|
dfss2 | ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss 3260 | . . 3 ⊢ (A ⊆ B ↔ A = (A ∩ B)) | |
2 | dfcleq 2347 | . . . 4 ⊢ (A = (A ∩ B) ↔ ∀x(x ∈ A ↔ x ∈ (A ∩ B))) | |
3 | elin 3219 | . . . . . 6 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B)) | |
4 | 3 | bibi2i 304 | . . . . 5 ⊢ ((x ∈ A ↔ x ∈ (A ∩ B)) ↔ (x ∈ A ↔ (x ∈ A ∧ x ∈ B))) |
5 | 4 | albii 1566 | . . . 4 ⊢ (∀x(x ∈ A ↔ x ∈ (A ∩ B)) ↔ ∀x(x ∈ A ↔ (x ∈ A ∧ x ∈ B))) |
6 | 2, 5 | bitri 240 | . . 3 ⊢ (A = (A ∩ B) ↔ ∀x(x ∈ A ↔ (x ∈ A ∧ x ∈ B))) |
7 | 1, 6 | bitri 240 | . 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A ↔ (x ∈ A ∧ x ∈ B))) |
8 | pm4.71 611 | . . 3 ⊢ ((x ∈ A → x ∈ B) ↔ (x ∈ A ↔ (x ∈ A ∧ x ∈ B))) | |
9 | 8 | albii 1566 | . 2 ⊢ (∀x(x ∈ A → x ∈ B) ↔ ∀x(x ∈ A ↔ (x ∈ A ∧ x ∈ B))) |
10 | 7, 9 | bitr4i 243 | 1 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∀wal 1540 = wceq 1642 ∈ wcel 1710 ∩ cin 3208 ⊆ wss 3257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 |
This theorem is referenced by: dfss3 3263 dfss2f 3264 ssel 3267 ssriv 3277 ssrdv 3278 sstr2 3279 eqss 3287 nss 3329 rabss2 3349 ssconb 3399 ssequn1 3433 unss 3437 ssin 3477 reldisj 3594 ssdif0 3609 difin0ss 3616 inssdif0 3617 ssundif 3633 sbcss 3660 sscon34 3661 pwss 3736 snss 3838 pwpw0 3855 pwsnALT 3882 disj5 3890 ssuni 3913 unissb 3921 intss 3947 iunss 4007 ssofss 4076 ssetkex 4294 dfpw2 4327 funimass4 5368 clos1induct 5880 dfnnc3 5885 ncssfin 6151 |
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