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| Mirrors > Home > MPE Home > Th. List > Mathboxes > scotteld | Structured version Visualization version GIF version | ||
| Description: The Scott operation sends inhabited classes to inhabited sets. (Contributed by Rohan Ridenour, 11-Aug-2023.) | 
| Ref | Expression | 
|---|---|
| scotteld.1 | ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) | 
| Ref | Expression | 
|---|---|
| scotteld | ⊢ (𝜑 → ∃𝑥 𝑥 ∈ Scott 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | scotteld.1 | . . . . . 6 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) | |
| 2 | n0 4352 | . . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
| 3 | 1, 2 | sylibr 234 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ ∅) | 
| 4 | 3 | neneqd 2944 | . . . 4 ⊢ (𝜑 → ¬ 𝐴 = ∅) | 
| 5 | scott0 9927 | . . . . 5 ⊢ (𝐴 = ∅ ↔ {𝑦 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)} = ∅) | |
| 6 | df-scott 44260 | . . . . . 6 ⊢ Scott 𝐴 = {𝑦 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)} | |
| 7 | 6 | eqeq1i 2741 | . . . . 5 ⊢ (Scott 𝐴 = ∅ ↔ {𝑦 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)} = ∅) | 
| 8 | 5, 7 | bitr4i 278 | . . . 4 ⊢ (𝐴 = ∅ ↔ Scott 𝐴 = ∅) | 
| 9 | 4, 8 | sylnib 328 | . . 3 ⊢ (𝜑 → ¬ Scott 𝐴 = ∅) | 
| 10 | 9 | neqned 2946 | . 2 ⊢ (𝜑 → Scott 𝐴 ≠ ∅) | 
| 11 | n0 4352 | . 2 ⊢ (Scott 𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ Scott 𝐴) | |
| 12 | 10, 11 | sylib 218 | 1 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ Scott 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 {crab 3435 ⊆ wss 3950 ∅c0 4332 ‘cfv 6560 rankcrnk 9804 Scott cscott 44259 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-r1 9805 df-rank 9806 df-scott 44260 | 
| This theorem is referenced by: cpcolld 44282 | 
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