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Theorem scott0f 33606
Description: A version of scott0 8693 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 8693 . 2 (𝐴 = ∅ ↔ {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = ∅)
2 scott0f.1 . . . . . . 7 𝑦𝐴
3 nfcv 2761 . . . . . . 7 𝑧𝐴
4 nfv 1840 . . . . . . 7 𝑧(rank‘𝑥) ⊆ (rank‘𝑦)
5 nfv 1840 . . . . . . 7 𝑦(rank‘𝑥) ⊆ (rank‘𝑧)
6 fveq2 6148 . . . . . . . 8 (𝑦 = 𝑧 → (rank‘𝑦) = (rank‘𝑧))
76sseq2d 3612 . . . . . . 7 (𝑦 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
82, 3, 4, 5, 7cbvralf 3153 . . . . . 6 (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧))
98a1i 11 . . . . 5 (𝑥𝐴 → (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)))
109rabbiia 3173 . . . 4 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥𝐴 ∣ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
11 nfcv 2761 . . . . 5 𝑤𝐴
12 scott0f.2 . . . . 5 𝑥𝐴
13 nfv 1840 . . . . . 6 𝑥(rank‘𝑤) ⊆ (rank‘𝑧)
1412, 13nfral 2940 . . . . 5 𝑥𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)
15 nfv 1840 . . . . 5 𝑤𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)
16 fveq2 6148 . . . . . . 7 (𝑤 = 𝑥 → (rank‘𝑤) = (rank‘𝑥))
1716sseq1d 3611 . . . . . 6 (𝑤 = 𝑥 → ((rank‘𝑤) ⊆ (rank‘𝑧) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
1817ralbidv 2980 . . . . 5 (𝑤 = 𝑥 → (∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) ↔ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)))
1911, 12, 14, 15, 18cbvrab 3184 . . . 4 {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = {𝑥𝐴 ∣ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
2010, 19eqtr4i 2646 . . 3 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)}
2120eqeq1i 2626 . 2 ({𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅ ↔ {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = ∅)
221, 21bitr4i 267 1 (𝐴 = ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wcel 1987  wnfc 2748  wral 2907  {crab 2911  wss 3555  c0 3891  cfv 5847  rankcrnk 8570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-r1 8571  df-rank 8572
This theorem is referenced by:  scottn0f  33607
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