Theorem 0elscott 35813

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 6890	. . . . . . . 8 ⊢ (𝑦 = ∅ → (rank‘𝑦) = (rank‘∅))
2	1	sseq2d 3941	. . . . . . 7 ⊢ (𝑦 = ∅ → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘∅)))
3		ss0b 4350	. . . . . . . 8 ⊢ ((rank‘𝑥) ⊆ ∅ ↔ (rank‘𝑥) = ∅)
4		0elon 6421	. . . . . . . . . 10 ⊢ ∅ ∈ On
5		onrankid 35785	. . . . . . . . . 10 ⊢ (∅ ∈ On ↔ (rank‘∅) = ∅)
6	4, 5	mpbi 233	. . . . . . . . 9 ⊢ (rank‘∅) = ∅
7	6	sseq2i 3938	. . . . . . . 8 ⊢ ((rank‘𝑥) ⊆ (rank‘∅) ↔ (rank‘𝑥) ⊆ ∅)
8		vex 3454	. . . . . . . . 9 ⊢ 𝑥 ∈ V
9	8	rankeq0 9884	. . . . . . . 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 3572	. . . . 5 ⊢ (∅ ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → 𝑥 = ∅))
13	12	ralrimivw 3158	. . . 4 ⊢ (∅ ∈ 𝐴 → ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → 𝑥 = ∅))
14		df-scott 9936	. . . . . 6 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
15	14	sseq1i 3937	. . . . 5 ⊢ (Scott 𝐴 ⊆ {∅} ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {∅})
16		rabsssn 4625	. . . . 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 4781	. . 3 ⊢ (Scott 𝐴 ⊆ {∅} ↔ (Scott 𝐴 = ∅ ∨ Scott 𝐴 = {∅}))
20	18, 19	sylib 221	. 2 ⊢ (∅ ∈ 𝐴 → (Scott 𝐴 = ∅ ∨ Scott 𝐴 = {∅}))
21		elscottn0 35812	. . 3 ⊢ (∅ ∈ 𝐴 → Scott 𝐴 ≠ ∅)
22	21	neneqd 2960	. 2 ⊢ (∅ ∈ 𝐴 → ¬ Scott 𝐴 = ∅)
23	20, 22	orcnd 892	1 ⊢ (∅ ∈ 𝐴 → Scott 𝐴 = {∅})

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412   ⊆ wss 3896  ∅c0 4277  {csn 4584  Oncon0 6365  ‘cfv 6543  rankcrnk 9774  Scott cscott 9935

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-reg 9593  ax-inf2 9649

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-om 7882  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-r1 9775  df-rank 9776  df-scott 9936

This theorem is used by: (None)