![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > unab | Structured version Visualization version GIF version |
Description: Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
unab | ⊢ ({𝑥 ∣ 𝜑} ∪ {𝑥 ∣ 𝜓}) = {𝑥 ∣ (𝜑 ∨ 𝜓)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbor 2296 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓)) | |
2 | df-clab 2703 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∨ 𝜓)} ↔ [𝑦 / 𝑥](𝜑 ∨ 𝜓)) | |
3 | df-clab 2703 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
4 | df-clab 2703 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓) | |
5 | 3, 4 | orbi12i 912 | . . 3 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ∨ 𝑦 ∈ {𝑥 ∣ 𝜓}) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓)) |
6 | 1, 2, 5 | 3bitr4ri 303 | . 2 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ∨ 𝑦 ∈ {𝑥 ∣ 𝜓}) ↔ 𝑦 ∈ {𝑥 ∣ (𝜑 ∨ 𝜓)}) |
7 | 6 | uneqri 4148 | 1 ⊢ ({𝑥 ∣ 𝜑} ∪ {𝑥 ∣ 𝜓}) = {𝑥 ∣ (𝜑 ∨ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 845 = wceq 1533 [wsb 2059 ∈ wcel 2098 {cab 2702 ∪ cun 3942 |
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-10 2129 ax-12 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3463 df-un 3949 |
This theorem is referenced by: unrab 4304 rabun2 4313 dmun 5913 hashf1lem2 14453 vdwlem6 16958 addsasslem1 27966 addsasslem2 27967 addsdilem1 28101 addsdilem2 28102 mulsasslem1 28113 mulsasslem2 28114 vtxdun 29367 satfvsuclem1 35100 satf0suclem 35116 fmlasuc0 35125 sticksstones22 41771 diophun 42335 |
Copyright terms: Public domain | W3C validator |