Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ss2abdv | Structured version Visualization version GIF version |
Description: Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.) |
Ref | Expression |
---|---|
ss2abdv.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
ss2abdv | ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ss2abdv.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 1 | alrimiv 1919 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒)) |
3 | ss2ab 4036 | . 2 ⊢ ({𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒} ↔ ∀𝑥(𝜓 → 𝜒)) | |
4 | 2, 3 | sylibr 235 | 1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1526 {cab 2796 ⊆ wss 3933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-in 3940 df-ss 3949 |
This theorem is referenced by: intss 4888 ssopab2 5424 ssoprab2 7211 suppimacnvss 7829 suppimacnv 7830 ressuppss 7838 ss2ixp 8462 fiss 8876 tcss 9174 tcel 9175 infmap2 9628 cfub 9659 cflm 9660 cflecard 9663 clsslem 14332 cncmet 23852 plyss 24716 iunrnmptss 30245 ofrn2 30315 sigaclci 31290 subfacp1lem6 32329 ss2mcls 32712 itg2addnclem 34824 sdclem1 34899 istotbnd3 34930 sstotbnd 34934 qsss1 35426 aomclem4 39535 hbtlem4 39604 hbtlem3 39605 rngunsnply 39651 iocinico 39696 |
Copyright terms: Public domain | W3C validator |