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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfon2lem2 | Structured version Visualization version GIF version |
Description: Lemma for dfon2 34406. (Contributed by Scott Fenton, 28-Feb-2011.) |
Ref | Expression |
---|---|
dfon2lem2 | ⊢ ∪ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1137 | . . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓) → 𝑥 ⊆ 𝐴) | |
2 | 1 | ss2abi 4028 | . . 3 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴} |
3 | df-pw 4567 | . . 3 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} | |
4 | 2, 3 | sseqtrri 3986 | . 2 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝒫 𝐴 |
5 | sspwuni 5065 | . 2 ⊢ ({𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝒫 𝐴 ↔ ∪ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝐴) | |
6 | 4, 5 | mpbi 229 | 1 ⊢ ∪ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1088 {cab 2714 ⊆ wss 3915 𝒫 cpw 4565 ∪ cuni 4870 |
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-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-3an 1090 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-v 3450 df-in 3922 df-ss 3932 df-pw 4567 df-uni 4871 |
This theorem is referenced by: dfon2lem3 34399 dfon2lem7 34403 |
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