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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 | rabid2 3381 | . . 3 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑)) | |
2 | exmidd 892 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑)) | |
3 | 1, 2 | mprgbir 3153 | . 2 ⊢ 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} |
4 | unrab 4274 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} | |
5 | 3, 4 | eqtr4i 2847 | 1 ⊢ 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 843 = wceq 1537 ∈ wcel 2114 {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-ral 3143 df-rab 3147 df-v 3496 df-un 3941 |
This theorem is referenced by: elnelun 4343 vtxdgoddnumeven 27335 esumrnmpt2 31327 ddemeas 31495 ballotth 31795 mbfposadd 34954 jm2.22 39612 |
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