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Theorem scott0f 38129
Description: A version of scott0 9955 with nonfree 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 9955 . 2 (𝐴 = ∅ ↔ {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = ∅)
2 scott0f.1 . . . . . 6 𝑦𝐴
3 nfcv 2908 . . . . . 6 𝑧𝐴
4 nfv 1913 . . . . . 6 𝑧(rank‘𝑥) ⊆ (rank‘𝑦)
5 nfv 1913 . . . . . 6 𝑦(rank‘𝑥) ⊆ (rank‘𝑧)
6 fveq2 6920 . . . . . . 7 (𝑦 = 𝑧 → (rank‘𝑦) = (rank‘𝑧))
76sseq2d 4041 . . . . . 6 (𝑦 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
82, 3, 4, 5, 7cbvralfw 3310 . . . . 5 (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧))
98rabbii 3449 . . . 4 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥𝐴 ∣ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
10 nfcv 2908 . . . . 5 𝑤𝐴
11 scott0f.2 . . . . 5 𝑥𝐴
12 nfv 1913 . . . . . 6 𝑥(rank‘𝑤) ⊆ (rank‘𝑧)
1311, 12nfralw 3317 . . . . 5 𝑥𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)
14 nfv 1913 . . . . 5 𝑤𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)
15 fveq2 6920 . . . . . . 7 (𝑤 = 𝑥 → (rank‘𝑤) = (rank‘𝑥))
1615sseq1d 4040 . . . . . 6 (𝑤 = 𝑥 → ((rank‘𝑤) ⊆ (rank‘𝑧) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
1716ralbidv 3184 . . . . 5 (𝑤 = 𝑥 → (∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) ↔ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)))
1810, 11, 13, 14, 17cbvrabw 3481 . . . 4 {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = {𝑥𝐴 ∣ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
199, 18eqtr4i 2771 . . 3 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)}
2019eqeq1i 2745 . 2 ({𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅ ↔ {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = ∅)
211, 20bitr4i 278 1 (𝐴 = ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wnfc 2893  wral 3067  {crab 3443  wss 3976  c0 4352  cfv 6573  rankcrnk 9832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-r1 9833  df-rank 9834
This theorem is referenced by:  scottn0f  38130
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