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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > scottrankd | Structured version Visualization version GIF version |
Description: Rank of a nonempty Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
Ref | Expression |
---|---|
scottrankd.1 | ⊢ (𝜑 → 𝐵 ∈ Scott 𝐴) |
Ref | Expression |
---|---|
scottrankd | ⊢ (𝜑 → (rank‘Scott 𝐴) = suc (rank‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scottex2 40953 | . . . 4 ⊢ Scott 𝐴 ∈ V | |
2 | 1 | rankval4 9280 | . . 3 ⊢ (rank‘Scott 𝐴) = ∪ 𝑥 ∈ Scott 𝐴 suc (rank‘𝑥) |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → (rank‘Scott 𝐴) = ∪ 𝑥 ∈ Scott 𝐴 suc (rank‘𝑥)) |
4 | scottrankd.1 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ Scott 𝐴) | |
5 | 4 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ Scott 𝐴) → 𝐵 ∈ Scott 𝐴) |
6 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ Scott 𝐴) → 𝑥 ∈ Scott 𝐴) | |
7 | 5, 6 | scottelrankd 40955 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ Scott 𝐴) → (rank‘𝐵) ⊆ (rank‘𝑥)) |
8 | 6, 5 | scottelrankd 40955 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ Scott 𝐴) → (rank‘𝑥) ⊆ (rank‘𝐵)) |
9 | 7, 8 | eqssd 3932 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ Scott 𝐴) → (rank‘𝐵) = (rank‘𝑥)) |
10 | 9 | suceqd 40917 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ Scott 𝐴) → suc (rank‘𝐵) = suc (rank‘𝑥)) |
11 | 10 | iuneq2dv 4905 | . 2 ⊢ (𝜑 → ∪ 𝑥 ∈ Scott 𝐴 suc (rank‘𝐵) = ∪ 𝑥 ∈ Scott 𝐴 suc (rank‘𝑥)) |
12 | 4 | ne0d 4251 | . . 3 ⊢ (𝜑 → Scott 𝐴 ≠ ∅) |
13 | iunconst 4890 | . . 3 ⊢ (Scott 𝐴 ≠ ∅ → ∪ 𝑥 ∈ Scott 𝐴 suc (rank‘𝐵) = suc (rank‘𝐵)) | |
14 | 12, 13 | syl 17 | . 2 ⊢ (𝜑 → ∪ 𝑥 ∈ Scott 𝐴 suc (rank‘𝐵) = suc (rank‘𝐵)) |
15 | 3, 11, 14 | 3eqtr2d 2839 | 1 ⊢ (𝜑 → (rank‘Scott 𝐴) = suc (rank‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 ∪ ciun 4881 suc csuc 6161 ‘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-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-reg 9040 ax-inf2 9088 |
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-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: gruscottcld 40957 |
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