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Mirrors > Home > MPE Home > Th. List > modom | Structured version Visualization version GIF version |
Description: Two ways to express "at most one". (Contributed by Stefan O'Rear, 28-Oct-2014.) |
Ref | Expression |
---|---|
modom | ⊢ (∃*𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≼ 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moeu 2602 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
2 | imor 839 | . 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑)) | |
3 | abn0 4223 | . . . . . 6 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑) | |
4 | 3 | necon1bbii 3017 | . . . . 5 ⊢ (¬ ∃𝑥𝜑 ↔ {𝑥 ∣ 𝜑} = ∅) |
5 | sdom1 8513 | . . . . 5 ⊢ ({𝑥 ∣ 𝜑} ≺ 1o ↔ {𝑥 ∣ 𝜑} = ∅) | |
6 | 4, 5 | bitr4i 270 | . . . 4 ⊢ (¬ ∃𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≺ 1o) |
7 | euen1 8376 | . . . 4 ⊢ (∃!𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≈ 1o) | |
8 | 6, 7 | orbi12i 898 | . . 3 ⊢ ((¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑) ↔ ({𝑥 ∣ 𝜑} ≺ 1o ∨ {𝑥 ∣ 𝜑} ≈ 1o)) |
9 | brdom2 8336 | . . 3 ⊢ ({𝑥 ∣ 𝜑} ≼ 1o ↔ ({𝑥 ∣ 𝜑} ≺ 1o ∨ {𝑥 ∣ 𝜑} ≈ 1o)) | |
10 | 8, 9 | bitr4i 270 | . 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑) ↔ {𝑥 ∣ 𝜑} ≼ 1o) |
11 | 1, 2, 10 | 3bitri 289 | 1 ⊢ (∃*𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≼ 1o) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∨ wo 833 = wceq 1507 ∃wex 1742 ∃*wmo 2545 ∃!weu 2583 {cab 2759 ∅c0 4179 class class class wbr 4929 1oc1o 7898 ≈ cen 8303 ≼ cdom 8304 ≺ csdm 8305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3418 df-sbc 3683 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-om 7397 df-1o 7905 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 |
This theorem is referenced by: modom2 8515 |
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