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Mirrors > Home > MPE Home > Th. List > ss2ab | Structured version Visualization version GIF version |
Description: Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.) |
Ref | Expression |
---|---|
ss2ab | ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfab1 2905 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
2 | nfab1 2905 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓} | |
3 | 1, 2 | dfss2f 3971 | . 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓})) |
4 | abid 2713 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
5 | abid 2713 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓) | |
6 | 4, 5 | imbi12i 350 | . . 3 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ (𝜑 → 𝜓)) |
7 | 6 | albii 1821 | . 2 ⊢ (∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ ∀𝑥(𝜑 → 𝜓)) |
8 | 3, 7 | bitri 274 | 1 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1539 ∈ wcel 2106 {cab 2709 ⊆ wss 3947 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-v 3476 df-in 3954 df-ss 3964 |
This theorem is referenced by: abss 4056 ssab 4057 ss2abdvOLD 4061 ss2abiOLD 4063 ss2rab 4067 rabss2 4074 rabsssn 4669 bj-gabss 35803 clss2lem 42347 ssabf 43774 abssf 43786 cfsetssfset 45752 sprssspr 46135 |
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