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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 2571 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
2 | imor 851 | . 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑)) | |
3 | abn0 4382 | . . . . . 6 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑) | |
4 | 3 | necon1bbii 2979 | . . . . 5 ⊢ (¬ ∃𝑥𝜑 ↔ {𝑥 ∣ 𝜑} = ∅) |
5 | sdom1 9267 | . . . . 5 ⊢ ({𝑥 ∣ 𝜑} ≺ 1o ↔ {𝑥 ∣ 𝜑} = ∅) | |
6 | 4, 5 | bitr4i 277 | . . . 4 ⊢ (¬ ∃𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≺ 1o) |
7 | euen1 9051 | . . . 4 ⊢ (∃!𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≈ 1o) | |
8 | 6, 7 | orbi12i 912 | . . 3 ⊢ ((¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑) ↔ ({𝑥 ∣ 𝜑} ≺ 1o ∨ {𝑥 ∣ 𝜑} ≈ 1o)) |
9 | brdom2 9003 | . . 3 ⊢ ({𝑥 ∣ 𝜑} ≼ 1o ↔ ({𝑥 ∣ 𝜑} ≺ 1o ∨ {𝑥 ∣ 𝜑} ≈ 1o)) | |
10 | 8, 9 | bitr4i 277 | . 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑) ↔ {𝑥 ∣ 𝜑} ≼ 1o) |
11 | 1, 2, 10 | 3bitri 296 | 1 ⊢ (∃*𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≼ 1o) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 845 = wceq 1533 ∃wex 1773 ∃*wmo 2526 ∃!weu 2556 {cab 2702 ∅c0 4322 class class class wbr 5149 1oc1o 8480 ≈ cen 8961 ≼ cdom 8962 ≺ csdm 8963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-1o 8487 df-en 8965 df-dom 8966 df-sdom 8967 |
This theorem is referenced by: modom2 9270 |
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