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| Mirrors > Home > MPE Home > Th. List > rabxm | Structured version Visualization version GIF version | ||
| Description: Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.) |
| Ref | Expression |
|---|---|
| rabxm | ⊢ 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rabid2im 3469 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑) → 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}) | |
| 2 | exmidd 896 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑)) | |
| 3 | 1, 2 | mprg 3067 | . 2 ⊢ 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} |
| 4 | unrab 4315 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} | |
| 5 | 3, 4 | eqtr4i 2768 | 1 ⊢ 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∨ wo 848 = wceq 1540 ∈ wcel 2108 {crab 3436 ∪ cun 3949 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-un 3956 |
| This theorem is referenced by: elnelun 4393 vtxdgoddnumeven 29571 esumrnmpt2 34069 ddemeas 34237 ballotth 34540 mbfposadd 37674 jm2.22 43007 |
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