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Theorem scottrankd 42620
Description: Rank of a nonempty Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Hypothesis
Ref Expression
scottrankd.1 (πœ‘ β†’ 𝐡 ∈ Scott 𝐴)
Assertion
Ref Expression
scottrankd (πœ‘ β†’ (rankβ€˜Scott 𝐴) = suc (rankβ€˜π΅))

Proof of Theorem scottrankd
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 scottex2 42617 . . . 4 Scott 𝐴 ∈ V
21rankval4 9811 . . 3 (rankβ€˜Scott 𝐴) = βˆͺ π‘₯ ∈ Scott 𝐴 suc (rankβ€˜π‘₯)
32a1i 11 . 2 (πœ‘ β†’ (rankβ€˜Scott 𝐴) = βˆͺ π‘₯ ∈ Scott 𝐴 suc (rankβ€˜π‘₯))
4 scottrankd.1 . . . . . . 7 (πœ‘ β†’ 𝐡 ∈ Scott 𝐴)
54adantr 482 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ Scott 𝐴) β†’ 𝐡 ∈ Scott 𝐴)
6 simpr 486 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ Scott 𝐴) β†’ π‘₯ ∈ Scott 𝐴)
75, 6scottelrankd 42619 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ Scott 𝐴) β†’ (rankβ€˜π΅) βŠ† (rankβ€˜π‘₯))
86, 5scottelrankd 42619 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ Scott 𝐴) β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π΅))
97, 8eqssd 3965 . . . 4 ((πœ‘ ∧ π‘₯ ∈ Scott 𝐴) β†’ (rankβ€˜π΅) = (rankβ€˜π‘₯))
109suceqd 42576 . . 3 ((πœ‘ ∧ π‘₯ ∈ Scott 𝐴) β†’ suc (rankβ€˜π΅) = suc (rankβ€˜π‘₯))
1110iuneq2dv 4982 . 2 (πœ‘ β†’ βˆͺ π‘₯ ∈ Scott 𝐴 suc (rankβ€˜π΅) = βˆͺ π‘₯ ∈ Scott 𝐴 suc (rankβ€˜π‘₯))
124ne0d 4299 . . 3 (πœ‘ β†’ Scott 𝐴 β‰  βˆ…)
13 iunconst 4967 . . 3 (Scott 𝐴 β‰  βˆ… β†’ βˆͺ π‘₯ ∈ Scott 𝐴 suc (rankβ€˜π΅) = suc (rankβ€˜π΅))
1412, 13syl 17 . 2 (πœ‘ β†’ βˆͺ π‘₯ ∈ Scott 𝐴 suc (rankβ€˜π΅) = suc (rankβ€˜π΅))
153, 11, 143eqtr2d 2779 1 (πœ‘ β†’ (rankβ€˜Scott 𝐴) = suc (rankβ€˜π΅))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ…c0 4286  βˆͺ ciun 4958  suc csuc 6323  β€˜cfv 6500  rankcrnk 9707  Scott cscott 42607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-reg 9536  ax-inf2 9585
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-r1 9708  df-rank 9709  df-scott 42608
This theorem is referenced by:  gruscottcld  42621
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