Theorem scott0f 38917

Description: A version of scott0b 9876 with nonfree variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)

Hypotheses

Ref	Expression
scott0f.1	⊢ Ⅎ𝑦𝐴
scott0f.2	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
scott0f	⊢ (𝐴 = ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (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.

Step	Hyp	Ref	Expression
1		df-scott 9868	. . 3 ⊢ Scott 𝐴 = {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)}
2	1	eqeq1i 2765	. 2 ⊢ (Scott 𝐴 = ∅ ↔ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = ∅)
3		scott0b 9876	. 2 ⊢ (𝐴 = ∅ ↔ Scott 𝐴 = ∅)
4		scott0f.1	. . . . . 6 ⊢ Ⅎ𝑦𝐴
5		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑧𝐴
6		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑧(rank‘𝑥) ⊆ (rank‘𝑦)
7		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑦(rank‘𝑥) ⊆ (rank‘𝑧)
8		fveq2 6878	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (rank‘𝑦) = (rank‘𝑧))
9	8	sseq2d 3963	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
10	4, 5, 6, 7, 9	cbvralfw 3302	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧))
11	10	rabbii 3417	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
12		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑤𝐴
13		scott0f.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
14		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑥(rank‘𝑤) ⊆ (rank‘𝑧)
15	13, 14	nfralw 3309	. . . . 5 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)
16		nfv 1947	. . . . 5 ⊢ Ⅎ𝑤∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)
17		fveq2 6878	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (rank‘𝑤) = (rank‘𝑥))
18	17	sseq1d 3962	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((rank‘𝑤) ⊆ (rank‘𝑧) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
19	18	ralbidv 3185	. . . . 5 ⊢ (𝑤 = 𝑥 → (∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) ↔ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)))
20	12, 13, 15, 16, 19	cbvrabw 3446	. . . 4 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = {𝑥 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
21	11, 20	eqtr4i 2786	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)}
22	21	eqeq1i 2765	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅ ↔ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = ∅)
23	2, 3, 22	3bitr4i 306	1 ⊢ (𝐴 = ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  Ⅎwnfc 2907  ∀wral 3076  {crab 3412   ⊆ wss 3899  ∅c0 4279  ‘cfv 6533  rankcrnk 9745  Scott cscott 9867

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-om 7863  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-r1 9746  df-rank 9747  df-scott 9868

This theorem is used by:  scottn0f  38918