![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dfss3f | Structured version Visualization version GIF version |
Description: Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.) |
Ref | Expression |
---|---|
dfss2f.1 | ⊢ Ⅎ𝑥𝐴 |
dfss2f.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
dfss3f | ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | dfss2f.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | dfss2f 3905 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
4 | df-ral 3111 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
5 | 3, 4 | bitr4i 281 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1536 ∈ wcel 2111 Ⅎwnfc 2936 ∀wral 3106 ⊆ wss 3881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-v 3443 df-in 3888 df-ss 3898 |
This theorem is referenced by: nfss 3907 sigaclcu2 31489 bnj1498 32443 heibor1 35248 ssrabf 41750 ssrab2f 41752 limsupequzmpt2 42360 liminfequzmpt2 42433 pimconstlt1 43340 pimltpnf 43341 pimiooltgt 43346 pimdecfgtioc 43350 pimincfltioc 43351 pimdecfgtioo 43352 pimincfltioo 43353 sssmf 43372 |
Copyright terms: Public domain | W3C validator |