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Theorem scott0f 35626
 Description: A version of scott0 9302 with non-free variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
Hypotheses
Ref Expression
scott0f.1 𝑦𝐴
scott0f.2 𝑥𝐴
Assertion
Ref Expression
scott0f (𝐴 = ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem scott0f
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 scott0 9302 . 2 (𝐴 = ∅ ↔ {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = ∅)
2 scott0f.1 . . . . . 6 𝑦𝐴
3 nfcv 2955 . . . . . 6 𝑧𝐴
4 nfv 1915 . . . . . 6 𝑧(rank‘𝑥) ⊆ (rank‘𝑦)
5 nfv 1915 . . . . . 6 𝑦(rank‘𝑥) ⊆ (rank‘𝑧)
6 fveq2 6646 . . . . . . 7 (𝑦 = 𝑧 → (rank‘𝑦) = (rank‘𝑧))
76sseq2d 3947 . . . . . 6 (𝑦 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
82, 3, 4, 5, 7cbvralfw 3382 . . . . 5 (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧))
98rabbii 3420 . . . 4 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥𝐴 ∣ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
10 nfcv 2955 . . . . 5 𝑤𝐴
11 scott0f.2 . . . . 5 𝑥𝐴
12 nfv 1915 . . . . . 6 𝑥(rank‘𝑤) ⊆ (rank‘𝑧)
1311, 12nfralw 3189 . . . . 5 𝑥𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)
14 nfv 1915 . . . . 5 𝑤𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)
15 fveq2 6646 . . . . . . 7 (𝑤 = 𝑥 → (rank‘𝑤) = (rank‘𝑥))
1615sseq1d 3946 . . . . . 6 (𝑤 = 𝑥 → ((rank‘𝑤) ⊆ (rank‘𝑧) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
1716ralbidv 3162 . . . . 5 (𝑤 = 𝑥 → (∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) ↔ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)))
1810, 11, 13, 14, 17cbvrabw 3437 . . . 4 {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = {𝑥𝐴 ∣ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
199, 18eqtr4i 2824 . . 3 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)}
2019eqeq1i 2803 . 2 ({𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅ ↔ {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = ∅)
211, 20bitr4i 281 1 (𝐴 = ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538  Ⅎwnfc 2936  ∀wral 3106  {crab 3110   ⊆ wss 3881  ∅c0 4243  ‘cfv 6325  rankcrnk 9179 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 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 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 4802  df-int 4840  df-iun 4884  df-iin 4885  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-om 7564  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-r1 9180  df-rank 9181 This theorem is referenced by:  scottn0f  35627
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