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Mirrors > Home > MPE Home > Th. List > Mathboxes > ss2ab1 | Structured version Visualization version GIF version |
Description: Class abstractions in a subclass relationship, closed form. One direction of ss2ab 4054 using fewer axioms. (Contributed by SN, 22-Dec-2024.) |
Ref | Expression |
---|---|
ss2ab1 | ⊢ (∀𝑥(𝜑 → 𝜓) → {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spsbim 2068 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓)) | |
2 | df-clab 2706 | . . 3 ⊢ (𝑡 ∈ {𝑥 ∣ 𝜑} ↔ [𝑡 / 𝑥]𝜑) | |
3 | df-clab 2706 | . . 3 ⊢ (𝑡 ∈ {𝑥 ∣ 𝜓} ↔ [𝑡 / 𝑥]𝜓) | |
4 | 1, 2, 3 | 3imtr4g 296 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝑡 ∈ {𝑥 ∣ 𝜑} → 𝑡 ∈ {𝑥 ∣ 𝜓})) |
5 | 4 | ssrdv 3986 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) → {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1532 [wsb 2060 ∈ wcel 2099 {cab 2705 ⊆ wss 3947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3473 df-in 3954 df-ss 3964 |
This theorem is referenced by: (None) |
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