Theorem wsuclem 32670

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 5990	. . . 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 5994	. . . . 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 3428	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ V)
9		wsuclem.4	. . . . 5 ⊢ (𝜑 → ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤)
10		rabn0 4219	. . . . . . 7 ⊢ ({𝑤 ∈ 𝐴 ∣ 𝑤◡𝑅𝑋} ≠ ∅ ↔ ∃𝑤 ∈ 𝐴 𝑤◡𝑅𝑋)
11		brcnvg 5596	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑋 ∈ V) → (𝑤◡𝑅𝑋 ↔ 𝑋𝑅𝑤))
12	11	ancoms 451	. . . . . . . 8 ⊢ ((𝑋 ∈ V ∧ 𝑤 ∈ 𝐴) → (𝑤◡𝑅𝑋 ↔ 𝑋𝑅𝑤))
13	12	rexbidva 3234	. . . . . . 7 ⊢ (𝑋 ∈ V → (∃𝑤 ∈ 𝐴 𝑤◡𝑅𝑋 ↔ ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤))
14	10, 13	syl5bb 275	. . . . . 6 ⊢ (𝑋 ∈ V → ({𝑤 ∈ 𝐴 ∣ 𝑤◡𝑅𝑋} ≠ ∅ ↔ ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤))
15	14	biimpar 470	. . . . 5 ⊢ ((𝑋 ∈ V ∧ ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤) → {𝑤 ∈ 𝐴 ∣ 𝑤◡𝑅𝑋} ≠ ∅)
16	8, 9, 15	syl2anc 576	. . . 4 ⊢ (𝜑 → {𝑤 ∈ 𝐴 ∣ 𝑤◡𝑅𝑋} ≠ ∅)
17	7, 16	eqnetrd 3027	. . 3 ⊢ (𝜑 → Pred(◡𝑅, 𝐴, 𝑋) ≠ ∅)
18		tz6.26 6014	. . 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 827	. 2 ⊢ (𝜑 → ∃𝑥 ∈ Pred (◡𝑅, 𝐴, 𝑋)Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅)
20		dfpred3g 5994	. . . . 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 3349	. . 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 4928	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦◡𝑅𝑋 ↔ 𝑥◡𝑅𝑋))
24	23	rexrab 3596	. . . 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 4177	. . . . . . . . . . . 12 ⊢ ¬ 𝑦 ∈ ∅
26		simp2r 1181	. . . . . . . . . . . . 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 2844	. . . . . . . . . . . 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 319	. . . . . . . . . . 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 3411	. . . . . . . . . . . . 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 1119	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋))
32		elpredg 5997	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → (𝑦 ∈ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) ↔ 𝑦𝑅𝑥))
33	30, 31, 32	syl2anc 576	. . . . . . . . . . 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 316	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → ¬ 𝑦𝑅𝑥)
35	34	3expa 1099	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅)) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → ¬ 𝑦𝑅𝑥)
36	35	ralrimiva 3125	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅)) → ∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑦𝑅𝑥)
37	36	expr 449	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅ → ∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑦𝑅𝑥))
38		simp1rl 1219	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐴)
39		simp1rr 1220	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑥◡𝑅𝑋)
40	5	adantr 473	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) → 𝑋 ∈ 𝑉)
41	40	3ad2ant1 1114	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑋 ∈ 𝑉)
42	29	elpred 5996	. . . . . . . . . . . . 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 701	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ Pred(◡𝑅, 𝐴, 𝑋))
45		simp3 1119	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑥𝑅𝑦)
46		breq1 4928	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑧𝑅𝑦 ↔ 𝑥𝑅𝑦))
47	46	rspcev 3528	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Pred(◡𝑅, 𝐴, 𝑋) ∧ 𝑥𝑅𝑦) → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦)
48	44, 45, 47	syl2anc 576	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦)
49	48	3expia 1102	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦))
50	49	ralrimiva 3125	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) → ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦))
51	50	expr 449	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥◡𝑅𝑋 → ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦)))
52	37, 51	anim12d 600	. . . . . 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 458	. . . . 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 3212	. . . 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 234	. . 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 232	. 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 198   ∧ wa 387   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051   ≠ wne 2960  ∀wral 3081  ∃wrex 3082  {crab 3085  Vcvv 3408   ⊆ wss 3822  ∅c0 4172   class class class wbr 4925   Se wse 5360   We wwe 5361  ^◡ccnv 5402  Predcpred 5982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-opab 4988  df-po 5322  df-so 5323  df-fr 5362  df-se 5363  df-we 5364  df-xp 5409  df-cnv 5411  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983

This theorem is referenced by:  wsuccl  32672  wsuclb  32673