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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 3462 | . . 3 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑)) | |
2 | exmidd 892 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑)) | |
3 | 1, 2 | mprgbir 3066 | . 2 ⊢ 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} |
4 | unrab 4306 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} | |
5 | 3, 4 | eqtr4i 2761 | 1 ⊢ 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 843 = wceq 1539 ∈ wcel 2104 {crab 3430 ∪ cun 3947 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-tru 1542 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ral 3060 df-rab 3431 df-v 3474 df-un 3954 |
This theorem is referenced by: elnelun 4390 vtxdgoddnumeven 29075 esumrnmpt2 33362 ddemeas 33530 ballotth 33832 mbfposadd 36840 jm2.22 42038 |
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