Theorem satf0suclem 35880

Description: Lemma for satf0suc 35881, sat1el2xp 35884 and fmlasuc0 35889. (Contributed by AV, 19-Sep-2023.)

Assertion

Ref	Expression
satf0suclem	⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶))} ∈ V)

Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑢,𝑈,𝑥,𝑦   𝑢,𝑉,𝑦   𝑢,𝑊,𝑦   𝑢,𝑋,𝑥,𝑦   𝑢,𝑌,𝑣,𝑥,𝑦   𝑢,𝑍,𝑤,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑦, 𝑤, 𝑣, 𝑢)   𝐶(𝑦, 𝑤, 𝑣, 𝑢)   𝑈(𝑤, 𝑣)   𝑉(𝑥, 𝑤, 𝑣)   𝑊(𝑥, 𝑤, 𝑣)   𝑋(𝑤, 𝑣)   𝑌(𝑤)   𝑍(𝑣)

Proof of Theorem satf0suclem

Step	Hyp	Ref	Expression
1		peano1 7887	. . . . . 6 ⊢ ∅ ∈ ω
2		eleq1 2853	. . . . . 6 ⊢ (𝑦 = ∅ → (𝑦 ∈ ω ↔ ∅ ∈ ω))
3	1, 2	mpbiri 261	. . . . 5 ⊢ (𝑦 = ∅ → 𝑦 ∈ ω)
4	3	adantr 486	. . . 4 ⊢ ((𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)) → 𝑦 ∈ ω)
5	4	pm4.71ri 570	. . 3 ⊢ ((𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)) ↔ (𝑦 ∈ ω ∧ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶))))
6	5	opabbii 5180	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶))} = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ ω ∧ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)))}
7		omex 9615	. . . . 5 ⊢ ω ∈ V
8	7	a1i 11	. . . 4 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ω ∈ V)
9		simp1 1154	. . . . . 6 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑋 ∈ 𝑈)
10		unab 4261	. . . . . . . 8 ⊢ ({𝑥 ∣ ∃𝑣 ∈ 𝑌 𝑥 = 𝐵} ∪ {𝑥 ∣ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶}) = {𝑥 ∣ (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)}
11		abrexexg 7960	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝑉 → {𝑥 ∣ ∃𝑣 ∈ 𝑌 𝑥 = 𝐵} ∈ V)
12	11	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑥 ∣ ∃𝑣 ∈ 𝑌 𝑥 = 𝐵} ∈ V)
13		abrexexg 7960	. . . . . . . . . 10 ⊢ (𝑍 ∈ 𝑊 → {𝑥 ∣ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶} ∈ V)
14	13	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑥 ∣ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶} ∈ V)
15		unexg 7747	. . . . . . . . 9 ⊢ (({𝑥 ∣ ∃𝑣 ∈ 𝑌 𝑥 = 𝐵} ∈ V ∧ {𝑥 ∣ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶} ∈ V) → ({𝑥 ∣ ∃𝑣 ∈ 𝑌 𝑥 = 𝐵} ∪ {𝑥 ∣ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶}) ∈ V)
16	12, 14, 15	syl2anc 596	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ({𝑥 ∣ ∃𝑣 ∈ 𝑌 𝑥 = 𝐵} ∪ {𝑥 ∣ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶}) ∈ V)
17	10, 16	eqeltrrid 2870	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑥 ∣ (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)} ∈ V)
18	17	ralrimivw 3163	. . . . . 6 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ∀𝑢 ∈ 𝑋 {𝑥 ∣ (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)} ∈ V)
19		abrexex2g 7963	. . . . . 6 ⊢ ((𝑋 ∈ 𝑈 ∧ ∀𝑢 ∈ 𝑋 {𝑥 ∣ (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)} ∈ V) → {𝑥 ∣ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)} ∈ V)
20	9, 18, 19	syl2anc 596	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑥 ∣ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)} ∈ V)
21	20	adantr 486	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑦 ∈ ω) → {𝑥 ∣ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)} ∈ V)
22	8, 21	opabex3rd 7965	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ ω ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶))} ∈ V)
23		simpr 490	. . . . . 6 ⊢ ((𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)) → ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶))
24	23	anim2i 629	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶))) → (𝑦 ∈ ω ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)))
25	24	ssopab2i 5537	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ ω ∧ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ ω ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶))}
26	25	a1i 11	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ ω ∧ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ ω ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶))})
27	22, 26	ssexd 5297	. 2 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ ω ∧ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)))} ∈ V)
28	6, 27	eqeltrid 2869	1 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶))} ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  {cab 2743  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ∪ cun 3904   ⊆ wss 3906  ∅c0 4286  {copab 5175  ωcom 7864

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-inf2 9613

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-tr 5221  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-om 7865

This theorem is used by:  satf0suc  35881  sat1el2xp  35884