wsuclem - Mathbox for Scott Fenton

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Theorem wsuclem 33746

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 6199	. . . 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 6203	. . . . 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 3442	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ V)
9		wsuclem.4	. . . . 5 ⊢ (𝜑 → ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤)
10		rabn0 4316	. . . . . . 7 ⊢ ({𝑤 ∈ 𝐴 ∣ 𝑤^◡𝑅𝑋} ≠ ∅ ↔ ∃𝑤 ∈ 𝐴 𝑤^◡𝑅𝑋)
11		brcnvg 5777	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑋 ∈ V) → (𝑤^◡𝑅𝑋 ↔ 𝑋𝑅𝑤))
12	11	ancoms 458	. . . . . . . 8 ⊢ ((𝑋 ∈ V ∧ 𝑤 ∈ 𝐴) → (𝑤^◡𝑅𝑋 ↔ 𝑋𝑅𝑤))
13	12	rexbidva 3224	. . . . . . 7 ⊢ (𝑋 ∈ V → (∃𝑤 ∈ 𝐴 𝑤^◡𝑅𝑋 ↔ ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤))
14	10, 13	syl5bb 282	. . . . . 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 583	. . . 4 ⊢ (𝜑 → {𝑤 ∈ 𝐴 ∣ 𝑤^◡𝑅𝑋} ≠ ∅)
17	7, 16	eqnetrd 3010	. . 3 ⊢ (𝜑 → Pred(^◡𝑅, 𝐴, 𝑋) ≠ ∅)
18		tz6.26 6235	. . 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 835	. 2 ⊢ (𝜑 → ∃𝑥 ∈ Pred (^◡𝑅, 𝐴, 𝑋)Pred(𝑅, Pred(^◡𝑅, 𝐴, 𝑋), 𝑥) = ∅)
20		dfpred3g 6203	. . . . 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 3340	. . 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 5073	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦^◡𝑅𝑋 ↔ 𝑥^◡𝑅𝑋))
24	23	rexrab 3626	. . . 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 4261	. . . . . . . . . . . 12 ⊢ ¬ 𝑦 ∈ ∅
26		simp2r 1198	. . . . . . . . . . . . 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 2824	. . . . . . . . . . . 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 326	. . . . . . . . . . 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 3426	. . . . . . . . . . . . 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 1136	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(^◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) ∧ 𝑦 ∈ Pred(^◡𝑅, 𝐴, 𝑋)) → 𝑦 ∈ Pred(^◡𝑅, 𝐴, 𝑋))
32		elpredg 6205	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ Pred(^◡𝑅, 𝐴, 𝑋)) → (𝑦 ∈ Pred(𝑅, Pred(^◡𝑅, 𝐴, 𝑋), 𝑥) ↔ 𝑦𝑅𝑥))
33	30, 31, 32	syl2anc 583	. . . . . . . . . . 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 323	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(^◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) ∧ 𝑦 ∈ Pred(^◡𝑅, 𝐴, 𝑋)) → ¬ 𝑦𝑅𝑥)
35	34	3expa 1116	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(^◡𝑅, 𝐴, 𝑋), 𝑥) = ∅)) ∧ 𝑦 ∈ Pred(^◡𝑅, 𝐴, 𝑋)) → ¬ 𝑦𝑅𝑥)
36	35	ralrimiva 3107	. . . . . . . 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 1236	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥^◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐴)
39		simp1rr 1237	. . . . . . . . . . . 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 1131	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥^◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑋 ∈ 𝑉)
42	29	elpred 6208	. . . . . . . . . . . . 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 709	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥^◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ Pred(^◡𝑅, 𝐴, 𝑋))
45		simp3 1136	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥^◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑥𝑅𝑦)
46		breq1 5073	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑧𝑅𝑦 ↔ 𝑥𝑅𝑦))
47	46	rspcev 3552	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Pred(^◡𝑅, 𝐴, 𝑋) ∧ 𝑥𝑅𝑦) → ∃𝑧 ∈ Pred (^◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦)
48	44, 45, 47	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥^◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → ∃𝑧 ∈ Pred (^◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦)
49	48	3expia 1119	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥^◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (^◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦))
50	49	ralrimiva 3107	. . . . . . . 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 608	. . . . . 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 3202	. . . 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	syl5bi 241	. . 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 239	. 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)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 {crab 3067 Vcvv 3422 ⊆ wss 3883 ∅c0 4253 class class class wbr 5070 Se wse 5533 We wwe 5534 ^◡ccnv 5579 Predcpred 6190

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-br 5071 df-opab 5133 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-cnv 5588 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191

This theorem is referenced by: wsuccl 33748 wsuclb 33749

W3C validator