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Mirrors > Home > NFE Home > Th. List > ssintab | GIF version |
Description: Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
ssintab | ⊢ (A ⊆ ∩{x ∣ φ} ↔ ∀x(φ → A ⊆ x)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssint 3942 | . 2 ⊢ (A ⊆ ∩{x ∣ φ} ↔ ∀y ∈ {x ∣ φ}A ⊆ y) | |
2 | sseq2 3293 | . . 3 ⊢ (y = x → (A ⊆ y ↔ A ⊆ x)) | |
3 | 2 | ralab2 3001 | . 2 ⊢ (∀y ∈ {x ∣ φ}A ⊆ y ↔ ∀x(φ → A ⊆ x)) |
4 | 1, 3 | bitri 240 | 1 ⊢ (A ⊆ ∩{x ∣ φ} ↔ ∀x(φ → A ⊆ x)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∀wal 1540 {cab 2339 ∀wral 2614 ⊆ wss 3257 ∩cint 3926 |
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 df-int 3927 |
This theorem is referenced by: ssmin 3945 ssintrab 3949 intmin4 3955 |
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