![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dfon2lem2 | Structured version Visualization version GIF version |
Description: Lemma for dfon2 35774. (Contributed by Scott Fenton, 28-Feb-2011.) |
Ref | Expression |
---|---|
dfon2lem2 | ⊢ ∪ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1135 | . . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓) → 𝑥 ⊆ 𝐴) | |
2 | 1 | ss2abi 4077 | . . 3 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴} |
3 | df-pw 4607 | . . 3 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} | |
4 | 2, 3 | sseqtrri 4033 | . 2 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝒫 𝐴 |
5 | sspwuni 5105 | . 2 ⊢ ({𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝒫 𝐴 ↔ ∪ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝐴) | |
6 | 4, 5 | mpbi 230 | 1 ⊢ ∪ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1086 {cab 2712 ⊆ wss 3963 𝒫 cpw 4605 ∪ cuni 4912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-v 3480 df-ss 3980 df-pw 4607 df-uni 4913 |
This theorem is referenced by: dfon2lem3 35767 dfon2lem7 35771 |
Copyright terms: Public domain | W3C validator |