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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 2753 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
2 | nfab1 2753 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓} | |
3 | 1, 2 | dfss2f 3559 | . 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓})) |
4 | abid 2598 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
5 | abid 2598 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓) | |
6 | 4, 5 | imbi12i 339 | . . 3 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ (𝜑 → 𝜓)) |
7 | 6 | albii 1737 | . 2 ⊢ (∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ ∀𝑥(𝜑 → 𝜓)) |
8 | 3, 7 | bitri 263 | 1 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 ∈ wcel 1977 {cab 2596 ⊆ wss 3540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-in 3547 df-ss 3554 |
This theorem is referenced by: abss 3634 ssab 3635 ss2abi 3637 ss2abdv 3638 ss2rab 3641 rabss2 3648 rabsssn 4162 clss2lem 36937 ssabf 38308 abssf 38326 |
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