Theorem wsuclem 35826

Description: Lemma for the supremum properties of well-founded successor. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)

Hypotheses

Ref	Expression
wsuclem.1	⊢ (𝜑 → 𝑅 We 𝐴)
wsuclem.2	⊢ (𝜑 → 𝑅 Se 𝐴)
wsuclem.3	⊢ (𝜑 → 𝑋 ∈ 𝑉)
wsuclem.4	⊢ (𝜑 → ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤)

Assertion

Ref	Expression
wsuclem	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦)))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧,𝑤   𝜑,𝑥,𝑦   𝑥,𝑅,𝑦,𝑧,𝑤   𝑥,𝑋,𝑦,𝑧,𝑤

Allowed substitution hints:   𝜑(𝑧,𝑤)   𝑉(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem wsuclem

Step	Hyp	Ref	Expression
1		wsuclem.1	. . 3 ⊢ (𝜑 → 𝑅 We 𝐴)
2		wsuclem.2	. . 3 ⊢ (𝜑 → 𝑅 Se 𝐴)
3		predss 6329	. . . 4 ⊢ Pred(◡𝑅, 𝐴, 𝑋) ⊆ 𝐴
4	3	a1i 11	. . 3 ⊢ (𝜑 → Pred(◡𝑅, 𝐴, 𝑋) ⊆ 𝐴)
5		wsuclem.3	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
6		dfpred3g 6333	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → Pred(◡𝑅, 𝐴, 𝑋) = {𝑤 ∈ 𝐴 ∣ 𝑤◡𝑅𝑋})
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → Pred(◡𝑅, 𝐴, 𝑋) = {𝑤 ∈ 𝐴 ∣ 𝑤◡𝑅𝑋})
8	5	elexd 3504	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ V)
9		wsuclem.4	. . . . 5 ⊢ (𝜑 → ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤)
10		rabn0 4389	. . . . . . 7 ⊢ ({𝑤 ∈ 𝐴 ∣ 𝑤◡𝑅𝑋} ≠ ∅ ↔ ∃𝑤 ∈ 𝐴 𝑤◡𝑅𝑋)
11		brcnvg 5890	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑋 ∈ V) → (𝑤◡𝑅𝑋 ↔ 𝑋𝑅𝑤))
12	11	ancoms 458	. . . . . . . 8 ⊢ ((𝑋 ∈ V ∧ 𝑤 ∈ 𝐴) → (𝑤◡𝑅𝑋 ↔ 𝑋𝑅𝑤))
13	12	rexbidva 3177	. . . . . . 7 ⊢ (𝑋 ∈ V → (∃𝑤 ∈ 𝐴 𝑤◡𝑅𝑋 ↔ ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤))
14	10, 13	bitrid 283	. . . . . 6 ⊢ (𝑋 ∈ V → ({𝑤 ∈ 𝐴 ∣ 𝑤◡𝑅𝑋} ≠ ∅ ↔ ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤))
15	14	biimpar 477	. . . . 5 ⊢ ((𝑋 ∈ V ∧ ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤) → {𝑤 ∈ 𝐴 ∣ 𝑤◡𝑅𝑋} ≠ ∅)
16	8, 9, 15	syl2anc 584	. . . 4 ⊢ (𝜑 → {𝑤 ∈ 𝐴 ∣ 𝑤◡𝑅𝑋} ≠ ∅)
17	7, 16	eqnetrd 3008	. . 3 ⊢ (𝜑 → Pred(◡𝑅, 𝐴, 𝑋) ≠ ∅)
18		tz6.26 6368	. . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (Pred(◡𝑅, 𝐴, 𝑋) ⊆ 𝐴 ∧ Pred(◡𝑅, 𝐴, 𝑋) ≠ ∅)) → ∃𝑥 ∈ Pred (◡𝑅, 𝐴, 𝑋)Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅)
19	1, 2, 4, 17, 18	syl22anc 839	. 2 ⊢ (𝜑 → ∃𝑥 ∈ Pred (◡𝑅, 𝐴, 𝑋)Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅)
20		dfpred3g 6333	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → Pred(◡𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦◡𝑅𝑋})
21	5, 20	syl 17	. . . 4 ⊢ (𝜑 → Pred(◡𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦◡𝑅𝑋})
22	21	rexeqdv 3327	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ Pred (◡𝑅, 𝐴, 𝑋)Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅ ↔ ∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝑦◡𝑅𝑋}Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅))
23		breq1 5146	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦◡𝑅𝑋 ↔ 𝑥◡𝑅𝑋))
24	23	rexrab 3702	. . . 4 ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝑦◡𝑅𝑋}Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐴 (𝑥◡𝑅𝑋 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅))
25		noel 4338	. . . . . . . . . . . 12 ⊢ ¬ 𝑦 ∈ ∅
26		simp2r 1201	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅)
27	26	eleq2d 2827	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → (𝑦 ∈ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) ↔ 𝑦 ∈ ∅))
28	25, 27	mtbiri 327	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → ¬ 𝑦 ∈ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥))
29		vex 3484	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
30	29	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → 𝑥 ∈ V)
31		simp3 1139	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋))
32		elpredg 6335	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → (𝑦 ∈ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) ↔ 𝑦𝑅𝑥))
33	30, 31, 32	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → (𝑦 ∈ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) ↔ 𝑦𝑅𝑥))
34	28, 33	mtbid 324	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → ¬ 𝑦𝑅𝑥)
35	34	3expa 1119	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅)) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → ¬ 𝑦𝑅𝑥)
36	35	ralrimiva 3146	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅)) → ∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑦𝑅𝑥)
37	36	expr 456	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅ → ∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑦𝑅𝑥))
38		simp1rl 1239	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐴)
39		simp1rr 1240	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑥◡𝑅𝑋)
40	5	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) → 𝑋 ∈ 𝑉)
41	40	3ad2ant1 1134	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑋 ∈ 𝑉)
42	29	elpred 6338	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝑉 → (𝑥 ∈ Pred(◡𝑅, 𝐴, 𝑋) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)))
43	41, 42	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → (𝑥 ∈ Pred(◡𝑅, 𝐴, 𝑋) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)))
44	38, 39, 43	mpbir2and 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ Pred(◡𝑅, 𝐴, 𝑋))
45		simp3 1139	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑥𝑅𝑦)
46		breq1 5146	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑧𝑅𝑦 ↔ 𝑥𝑅𝑦))
47	46	rspcev 3622	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Pred(◡𝑅, 𝐴, 𝑋) ∧ 𝑥𝑅𝑦) → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦)
48	44, 45, 47	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦)
49	48	3expia 1122	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦))
50	49	ralrimiva 3146	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) → ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦))
51	50	expr 456	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥◡𝑅𝑋 → ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦)))
52	37, 51	anim12d 609	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅ ∧ 𝑥◡𝑅𝑋) → (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦))))
53	52	ancomsd 465	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥◡𝑅𝑋 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) → (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦))))
54	53	reximdva 3168	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 (𝑥◡𝑅𝑋 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦))))
55	24, 54	biimtrid 242	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝑦◡𝑅𝑋}Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅ → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦))))
56	22, 55	sylbid 240	. 2 ⊢ (𝜑 → (∃𝑥 ∈ Pred (◡𝑅, 𝐴, 𝑋)Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅ → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦))))
57	19, 56	mpd 15	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3436  Vcvv 3480   ⊆ wss 3951  ∅c0 4333   class class class wbr 5143   Se wse 5635   We wwe 5636  ^◡ccnv 5684  Predcpred 6320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321

This theorem is referenced by:  wsuccl  35828  wsuclb  35829