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Theorem scottelrankd 42996
Description: Property of a Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Hypotheses
Ref Expression
scottelrankd.1 (πœ‘ β†’ 𝐡 ∈ Scott 𝐴)
scottelrankd.2 (πœ‘ β†’ 𝐢 ∈ Scott 𝐴)
Assertion
Ref Expression
scottelrankd (πœ‘ β†’ (rankβ€˜π΅) βŠ† (rankβ€˜πΆ))

Proof of Theorem scottelrankd
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6891 . . 3 (𝑦 = 𝐢 β†’ (rankβ€˜π‘¦) = (rankβ€˜πΆ))
21sseq2d 4014 . 2 (𝑦 = 𝐢 β†’ ((rankβ€˜π΅) βŠ† (rankβ€˜π‘¦) ↔ (rankβ€˜π΅) βŠ† (rankβ€˜πΆ)))
3 scottelrankd.1 . . . . 5 (πœ‘ β†’ 𝐡 ∈ Scott 𝐴)
4 df-scott 42985 . . . . 5 Scott 𝐴 = {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)}
53, 4eleqtrdi 2843 . . . 4 (πœ‘ β†’ 𝐡 ∈ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)})
6 fveq2 6891 . . . . . . 7 (π‘₯ = 𝐡 β†’ (rankβ€˜π‘₯) = (rankβ€˜π΅))
76sseq1d 4013 . . . . . 6 (π‘₯ = 𝐡 β†’ ((rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦) ↔ (rankβ€˜π΅) βŠ† (rankβ€˜π‘¦)))
87ralbidv 3177 . . . . 5 (π‘₯ = 𝐡 β†’ (βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦) ↔ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π΅) βŠ† (rankβ€˜π‘¦)))
98elrab 3683 . . . 4 (𝐡 ∈ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} ↔ (𝐡 ∈ 𝐴 ∧ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π΅) βŠ† (rankβ€˜π‘¦)))
105, 9sylib 217 . . 3 (πœ‘ β†’ (𝐡 ∈ 𝐴 ∧ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π΅) βŠ† (rankβ€˜π‘¦)))
1110simprd 496 . 2 (πœ‘ β†’ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π΅) βŠ† (rankβ€˜π‘¦))
12 scottelrankd.2 . . . 4 (πœ‘ β†’ 𝐢 ∈ Scott 𝐴)
1312, 4eleqtrdi 2843 . . 3 (πœ‘ β†’ 𝐢 ∈ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)})
14 elrabi 3677 . . 3 (𝐢 ∈ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} β†’ 𝐢 ∈ 𝐴)
1513, 14syl 17 . 2 (πœ‘ β†’ 𝐢 ∈ 𝐴)
162, 11, 15rspcdva 3613 1 (πœ‘ β†’ (rankβ€˜π΅) βŠ† (rankβ€˜πΆ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432   βŠ† wss 3948  β€˜cfv 6543  rankcrnk 9757  Scott cscott 42984
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-scott 42985
This theorem is referenced by:  scottrankd  42997
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