Theorem erscottd 35819

Description: Applying the Scott operation to equivalence classes preserves their equality property. (Contributed by BTernaryTau, 20-Oct-2026.)

Hypotheses

Ref	Expression
erscottd.1	⊢ (𝜑 → 𝑅 Er 𝑉)
erscottd.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
erscottd	⊢ (𝜑 → (𝐴𝑅𝐵 ↔ Scott {𝑥 ∣ 𝑥𝑅𝐴} = Scott {𝑥 ∣ 𝑥𝑅𝐵}))

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem erscottd

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		erscottd.1	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Er 𝑉)
2	1	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵 ∧ 𝑥𝑅𝐴) → 𝑅 Er 𝑉)
3		simp3 1156	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵 ∧ 𝑥𝑅𝐴) → 𝑥𝑅𝐴)
4		simp2 1155	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵 ∧ 𝑥𝑅𝐴) → 𝐴𝑅𝐵)
5	2, 3, 4	ertrd 8742	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵 ∧ 𝑥𝑅𝐴) → 𝑥𝑅𝐵)
6	5	3expia 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝑥𝑅𝐴 → 𝑥𝑅𝐵))
7	1	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵 ∧ 𝑥𝑅𝐵) → 𝑅 Er 𝑉)
8		simp3 1156	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵 ∧ 𝑥𝑅𝐵) → 𝑥𝑅𝐵)
9		simp2 1155	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵 ∧ 𝑥𝑅𝐵) → 𝐴𝑅𝐵)
10	7, 8, 9	ertr4d 8745	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵 ∧ 𝑥𝑅𝐵) → 𝑥𝑅𝐴)
11	10	3expia 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝑥𝑅𝐵 → 𝑥𝑅𝐴))
12	6, 11	impbid 215	. . . 4 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝑥𝑅𝐴 ↔ 𝑥𝑅𝐵))
13	12	abbidv 2826	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → {𝑥 ∣ 𝑥𝑅𝐴} = {𝑥 ∣ 𝑥𝑅𝐵})
14	13	scotteqd 9937	. 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → Scott {𝑥 ∣ 𝑥𝑅𝐴} = Scott {𝑥 ∣ 𝑥𝑅𝐵})
15		erscottd.2	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
16	1, 15	erref 8746	. . . . . . 7 ⊢ (𝜑 → 𝐴𝑅𝐴)
17		breq1 5105	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝐴 ↔ 𝐴𝑅𝐴))
18	15, 16, 17	elabd 3634	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝑥𝑅𝐴})
19		n0i 4285	. . . . . 6 ⊢ (𝐴 ∈ {𝑥 ∣ 𝑥𝑅𝐴} → ¬ {𝑥 ∣ 𝑥𝑅𝐴} = ∅)
20	18, 19	syl 18	. . . . 5 ⊢ (𝜑 → ¬ {𝑥 ∣ 𝑥𝑅𝐴} = ∅)
21		scott0b 9944	. . . . 5 ⊢ ({𝑥 ∣ 𝑥𝑅𝐴} = ∅ ↔ Scott {𝑥 ∣ 𝑥𝑅𝐴} = ∅)
22	20, 21	sylnib 331	. . . 4 ⊢ (𝜑 → ¬ Scott {𝑥 ∣ 𝑥𝑅𝐴} = ∅)
23		disjeq0 4408	. . . . . . 7 ⊢ ((Scott {𝑥 ∣ 𝑥𝑅𝐴} ∩ Scott {𝑥 ∣ 𝑥𝑅𝐵}) = ∅ → (Scott {𝑥 ∣ 𝑥𝑅𝐴} = Scott {𝑥 ∣ 𝑥𝑅𝐵} ↔ (Scott {𝑥 ∣ 𝑥𝑅𝐴} = ∅ ∧ Scott {𝑥 ∣ 𝑥𝑅𝐵} = ∅)))
24	23	biimpcd 252	. . . . . 6 ⊢ (Scott {𝑥 ∣ 𝑥𝑅𝐴} = Scott {𝑥 ∣ 𝑥𝑅𝐵} → ((Scott {𝑥 ∣ 𝑥𝑅𝐴} ∩ Scott {𝑥 ∣ 𝑥𝑅𝐵}) = ∅ → (Scott {𝑥 ∣ 𝑥𝑅𝐴} = ∅ ∧ Scott {𝑥 ∣ 𝑥𝑅𝐵} = ∅)))
25		simpl 488	. . . . . . 7 ⊢ ((Scott {𝑥 ∣ 𝑥𝑅𝐴} = ∅ ∧ Scott {𝑥 ∣ 𝑥𝑅𝐵} = ∅) → Scott {𝑥 ∣ 𝑥𝑅𝐴} = ∅)
26	25	con3i 155	. . . . . 6 ⊢ (¬ Scott {𝑥 ∣ 𝑥𝑅𝐴} = ∅ → ¬ (Scott {𝑥 ∣ 𝑥𝑅𝐴} = ∅ ∧ Scott {𝑥 ∣ 𝑥𝑅𝐵} = ∅))
27	24, 26	nsyli 158	. . . . 5 ⊢ (Scott {𝑥 ∣ 𝑥𝑅𝐴} = Scott {𝑥 ∣ 𝑥𝑅𝐵} → (¬ Scott {𝑥 ∣ 𝑥𝑅𝐴} = ∅ → ¬ (Scott {𝑥 ∣ 𝑥𝑅𝐴} ∩ Scott {𝑥 ∣ 𝑥𝑅𝐵}) = ∅))
28		df-ne 2956	. . . . . 6 ⊢ ((Scott {𝑥 ∣ 𝑥𝑅𝐴} ∩ Scott {𝑥 ∣ 𝑥𝑅𝐵}) ≠ ∅ ↔ ¬ (Scott {𝑥 ∣ 𝑥𝑅𝐴} ∩ Scott {𝑥 ∣ 𝑥𝑅𝐵}) = ∅)
29		ndisj 4317	. . . . . 6 ⊢ ((Scott {𝑥 ∣ 𝑥𝑅𝐴} ∩ Scott {𝑥 ∣ 𝑥𝑅𝐵}) ≠ ∅ ↔ ∃𝑦(𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐴} ∧ 𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐵}))
30	28, 29	bitr3i 280	. . . . 5 ⊢ (¬ (Scott {𝑥 ∣ 𝑥𝑅𝐴} ∩ Scott {𝑥 ∣ 𝑥𝑅𝐵}) = ∅ ↔ ∃𝑦(𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐴} ∧ 𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐵}))
31	27, 30	imbitrdi 254	. . . 4 ⊢ (Scott {𝑥 ∣ 𝑥𝑅𝐴} = Scott {𝑥 ∣ 𝑥𝑅𝐵} → (¬ Scott {𝑥 ∣ 𝑥𝑅𝐴} = ∅ → ∃𝑦(𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐴} ∧ 𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐵})))
32	22, 31	mpan9 516	. . 3 ⊢ ((𝜑 ∧ Scott {𝑥 ∣ 𝑥𝑅𝐴} = Scott {𝑥 ∣ 𝑥𝑅𝐵}) → ∃𝑦(𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐴} ∧ 𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐵}))
33	1	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐴} ∧ 𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐵}) → 𝑅 Er 𝑉)
34		breq1 5105	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥𝑅𝐴 ↔ 𝑦𝑅𝐴))
35	34	elscottab 9949	. . . . . . . 8 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐴} → 𝑦𝑅𝐴)
36	35	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐴} ∧ 𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐵}) → 𝑦𝑅𝐴)
37		breq1 5105	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥𝑅𝐵 ↔ 𝑦𝑅𝐵))
38	37	elscottab 9949	. . . . . . . 8 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐵} → 𝑦𝑅𝐵)
39	38	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐴} ∧ 𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐵}) → 𝑦𝑅𝐵)
40	33, 36, 39	ertr3d 8744	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐴} ∧ 𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐵}) → 𝐴𝑅𝐵)
41	40	3expib 1140	. . . . 5 ⊢ (𝜑 → ((𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐴} ∧ 𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐵}) → 𝐴𝑅𝐵))
42	41	exlimdv 1965	. . . 4 ⊢ (𝜑 → (∃𝑦(𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐴} ∧ 𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐵}) → 𝐴𝑅𝐵))
43	42	adantr 486	. . 3 ⊢ ((𝜑 ∧ Scott {𝑥 ∣ 𝑥𝑅𝐴} = Scott {𝑥 ∣ 𝑥𝑅𝐵}) → (∃𝑦(𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐴} ∧ 𝑦 ∈ Scott {𝑥 ∣ 𝑥𝑅𝐵}) → 𝐴𝑅𝐵))
44	32, 43	mpd 16	. 2 ⊢ ((𝜑 ∧ Scott {𝑥 ∣ 𝑥𝑅𝐴} = Scott {𝑥 ∣ 𝑥𝑅𝐵}) → 𝐴𝑅𝐵)
45	14, 44	impbida 813	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ Scott {𝑥 ∣ 𝑥𝑅𝐴} = Scott {𝑥 ∣ 𝑥𝑅𝐵}))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738   ≠ wne 2955   ∩ cin 4088  ∅c0 4277   class class class wbr 5102   Er wer 8722  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-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

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-er 8725  df-r1 9775  df-rank 9776  df-scott 9936

This theorem is used by:  ecscottd  35820