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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 3973 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
4 | df-ral 3063 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
5 | 3, 4 | bitr4i 278 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 ∈ wcel 2107 Ⅎwnfc 2884 ∀wral 3062 ⊆ wss 3949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-v 3477 df-in 3956 df-ss 3966 |
This theorem is referenced by: nfss 3975 sigaclcu2 33149 bnj1498 34103 heibor1 36726 ssrabf 43851 ssrab2f 43854 limsupequzmpt2 44482 liminfequzmpt2 44555 pimconstlt1 45466 pimltpnff 45467 pimiooltgt 45474 pimdecfgtioc 45479 pimincfltioc 45480 pimdecfgtioo 45481 pimincfltioo 45482 pimgtmnff 45486 sssmf 45502 |
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