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Mirrors > Home > MPE Home > Th. List > unrab | Structured version Visualization version GIF version |
Description: Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.) |
Ref | Expression |
---|---|
unrab | ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3147 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | df-rab 3147 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} | |
3 | 1, 2 | uneq12i 4137 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ 𝜓}) = ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∪ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) |
4 | df-rab 3147 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∨ 𝜓))} | |
5 | unab 4270 | . . . 4 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∪ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜓))} | |
6 | andi 1004 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝜑 ∨ 𝜓)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜓))) | |
7 | 6 | abbii 2886 | . . . 4 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∨ 𝜓))} = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜓))} |
8 | 5, 7 | eqtr4i 2847 | . . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∪ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∨ 𝜓))} |
9 | 4, 8 | eqtr4i 2847 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓)} = ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∪ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) |
10 | 3, 9 | eqtr4i 2847 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 {cab 2799 {crab 3142 ∪ cun 3934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-un 3941 |
This theorem is referenced by: rabxm 4340 kmlem3 9578 hashbclem 13811 phiprmpw 16113 efgsfo 18865 dsmmacl 20885 rrxmvallem 24007 mumul 25758 ppiub 25780 lgsquadlem2 25957 edglnl 26928 numclwwlk3lem2lem 28162 hasheuni 31344 measvuni 31473 aean 31503 subfacp1lem6 32432 lineunray 33608 cnambfre 34955 itg2addnclem2 34959 iblabsnclem 34970 orrabdioph 39398 undisjrab 40658 mndpsuppss 44439 |
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