Theorem 0elscott 35878

Description: If the empty set is an element of the input class, then the Scott's trick set is the singleton of the empty set. (Contributed by BTernaryTau, 23-Aug-2026.)

Assertion

Ref	Expression
0elscott	⊢ (∅ ∈ 𝐴 → Scott 𝐴 = {∅})

Proof of Theorem 0elscott

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6916	. . . . . . . 8 ⊢ (𝑦 = ∅ → (rank‘𝑦) = (rank‘∅))
2	1	sseq2d 3945	. . . . . . 7 ⊢ (𝑦 = ∅ → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘∅)))
3		ss0b 4354	. . . . . . . 8 ⊢ ((rank‘𝑥) ⊆ ∅ ↔ (rank‘𝑥) = ∅)
4		0elon 6444	. . . . . . . . . 10 ⊢ ∅ ∈ On
5		onrankid 35852	. . . . . . . . . 10 ⊢ (∅ ∈ On ↔ (rank‘∅) = ∅)
6	4, 5	mpbi 233	. . . . . . . . 9 ⊢ (rank‘∅) = ∅
7	6	sseq2i 3942	. . . . . . . 8 ⊢ ((rank‘𝑥) ⊆ (rank‘∅) ↔ (rank‘𝑥) ⊆ ∅)
8		vex 3458	. . . . . . . . 9 ⊢ 𝑥 ∈ V
9	8	rankeq0 9914	. . . . . . . 8 ⊢ (𝑥 = ∅ ↔ (rank‘𝑥) = ∅)
10	3, 7, 9	3bitr4i 306	. . . . . . 7 ⊢ ((rank‘𝑥) ⊆ (rank‘∅) ↔ 𝑥 = ∅)
11	2, 10	bitrdi 290	. . . . . 6 ⊢ (𝑦 = ∅ → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ 𝑥 = ∅))
12	11	rspcv 3576	. . . . 5 ⊢ (∅ ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → 𝑥 = ∅))
13	12	ralrimivw 3162	. . . 4 ⊢ (∅ ∈ 𝐴 → ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → 𝑥 = ∅))
14		df-scott 9966	. . . . . 6 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
15	14	sseq1i 3941	. . . . 5 ⊢ (Scott 𝐴 ⊆ {∅} ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {∅})
16		rabsssn 4630	. . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → 𝑥 = ∅))
17	15, 16	bitri 278	. . . 4 ⊢ (Scott 𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → 𝑥 = ∅))
18	13, 17	sylibr 237	. . 3 ⊢ (∅ ∈ 𝐴 → Scott 𝐴 ⊆ {∅})
19		sssn 4787	. . 3 ⊢ (Scott 𝐴 ⊆ {∅} ↔ (Scott 𝐴 = ∅ ∨ Scott 𝐴 = {∅}))
20	18, 19	sylib 221	. 2 ⊢ (∅ ∈ 𝐴 → (Scott 𝐴 = ∅ ∨ Scott 𝐴 = {∅}))
21		elscottn0 35877	. . 3 ⊢ (∅ ∈ 𝐴 → Scott 𝐴 ≠ ∅)
22	21	neneqd 2964	. 2 ⊢ (∅ ∈ 𝐴 → ¬ Scott 𝐴 = ∅)
23	20, 22	orcnd 892	1 ⊢ (∅ ∈ 𝐴 → Scott 𝐴 = {∅})

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416   ⊆ wss 3900  ∅c0 4281  {csn 4588  Oncon0 6388  ‘cfv 6566  rankcrnk 9804  Scott cscott 9965

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-reg 9623  ax-inf2 9679

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-r1 9805  df-rank 9806  df-scott 9966

This theorem is used by: (None)