![]() |
Mathbox for Rohan Ridenour |
< Previous
Next >
Nearby theorems |
|
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 4260 | . . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
3 | 1, 2 | sylibr 237 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ ∅) |
4 | 3 | neneqd 2992 | . . . 4 ⊢ (𝜑 → ¬ 𝐴 = ∅) |
5 | scott0 9299 | . . . . 5 ⊢ (𝐴 = ∅ ↔ {𝑦 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)} = ∅) | |
6 | df-scott 40944 | . . . . . 6 ⊢ Scott 𝐴 = {𝑦 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)} | |
7 | 6 | eqeq1i 2803 | . . . . 5 ⊢ (Scott 𝐴 = ∅ ↔ {𝑦 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)} = ∅) |
8 | 5, 7 | bitr4i 281 | . . . 4 ⊢ (𝐴 = ∅ ↔ Scott 𝐴 = ∅) |
9 | 4, 8 | sylnib 331 | . . 3 ⊢ (𝜑 → ¬ Scott 𝐴 = ∅) |
10 | 9 | neqned 2994 | . 2 ⊢ (𝜑 → Scott 𝐴 ≠ ∅) |
11 | n0 4260 | . 2 ⊢ (Scott 𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ Scott 𝐴) | |
12 | 10, 11 | sylib 221 | 1 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ Scott 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 {crab 3110 ⊆ wss 3881 ∅c0 4243 ‘cfv 6324 rankcrnk 9176 Scott cscott 40943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-r1 9177 df-rank 9178 df-scott 40944 |
This theorem is referenced by: cpcolld 40966 |
Copyright terms: Public domain | W3C validator |