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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfon2lem2 | Structured version Visualization version GIF version |
Description: Lemma for dfon2 35445. (Contributed by Scott Fenton, 28-Feb-2011.) |
Ref | Expression |
---|---|
dfon2lem2 | ⊢ ∪ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1133 | . . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓) → 𝑥 ⊆ 𝐴) | |
2 | 1 | ss2abi 4055 | . . 3 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴} |
3 | df-pw 4600 | . . 3 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} | |
4 | 2, 3 | sseqtrri 4010 | . 2 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝒫 𝐴 |
5 | sspwuni 5098 | . 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 1084 {cab 2702 ⊆ wss 3939 𝒫 cpw 4598 ∪ cuni 4903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-3an 1086 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-v 3465 df-ss 3956 df-pw 4600 df-uni 4904 |
This theorem is referenced by: dfon2lem3 35438 dfon2lem7 35442 |
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