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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 4310 | . . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
3 | 1, 2 | sylibr 236 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ ∅) |
4 | 3 | neneqd 3021 | . . . 4 ⊢ (𝜑 → ¬ 𝐴 = ∅) |
5 | scott0 9315 | . . . . 5 ⊢ (𝐴 = ∅ ↔ {𝑦 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)} = ∅) | |
6 | df-scott 40621 | . . . . . 6 ⊢ Scott 𝐴 = {𝑦 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)} | |
7 | 6 | eqeq1i 2826 | . . . . 5 ⊢ (Scott 𝐴 = ∅ ↔ {𝑦 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)} = ∅) |
8 | 5, 7 | bitr4i 280 | . . . 4 ⊢ (𝐴 = ∅ ↔ Scott 𝐴 = ∅) |
9 | 4, 8 | sylnib 330 | . . 3 ⊢ (𝜑 → ¬ Scott 𝐴 = ∅) |
10 | 9 | neqned 3023 | . 2 ⊢ (𝜑 → Scott 𝐴 ≠ ∅) |
11 | n0 4310 | . 2 ⊢ (Scott 𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ Scott 𝐴) | |
12 | 10, 11 | sylib 220 | 1 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ Scott 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 {crab 3142 ⊆ wss 3936 ∅c0 4291 ‘cfv 6355 rankcrnk 9192 Scott cscott 40620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-r1 9193 df-rank 9194 df-scott 40621 |
This theorem is referenced by: cpcolld 40643 |
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