Theorem satf0suclem 35442

Description: Lemma for satf0suc 35443, sat1el2xp 35446 and fmlasuc0 35451. (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 7827	. . . . . 6 ⊢ ∅ ∈ ω
2		eleq1 2821	. . . . . 6 ⊢ (𝑦 = ∅ → (𝑦 ∈ ω ↔ ∅ ∈ ω))
3	1, 2	mpbiri 258	. . . . 5 ⊢ (𝑦 = ∅ → 𝑦 ∈ ω)
4	3	adantr 480	. . . 4 ⊢ ((𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)) → 𝑦 ∈ ω)
5	4	pm4.71ri 560	. . 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 5162	. 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 9542	. . . . 5 ⊢ ω ∈ V
8	7	a1i 11	. . . 4 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ω ∈ V)
9		simp1 1136	. . . . . 6 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑋 ∈ 𝑈)
10		unab 4257	. . . . . . . 8 ⊢ ({𝑥 ∣ ∃𝑣 ∈ 𝑌 𝑥 = 𝐵} ∪ {𝑥 ∣ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶}) = {𝑥 ∣ (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)}
11		abrexexg 7901	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝑉 → {𝑥 ∣ ∃𝑣 ∈ 𝑌 𝑥 = 𝐵} ∈ V)
12	11	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑥 ∣ ∃𝑣 ∈ 𝑌 𝑥 = 𝐵} ∈ V)
13		abrexexg 7901	. . . . . . . . . 10 ⊢ (𝑍 ∈ 𝑊 → {𝑥 ∣ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶} ∈ V)
14	13	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑥 ∣ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶} ∈ V)
15		unexg 7684	. . . . . . . . 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 584	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ({𝑥 ∣ ∃𝑣 ∈ 𝑌 𝑥 = 𝐵} ∪ {𝑥 ∣ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶}) ∈ V)
17	10, 16	eqeltrrid 2838	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑥 ∣ (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)} ∈ V)
18	17	ralrimivw 3129	. . . . . 6 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ∀𝑢 ∈ 𝑋 {𝑥 ∣ (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)} ∈ V)
19		abrexex2g 7904	. . . . . 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 584	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑥 ∣ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)} ∈ V)
21	20	adantr 480	. . . 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 7906	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ ω ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶))} ∈ V)
23		simpr 484	. . . . . 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 617	. . . . 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 5495	. . . 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 5266	. 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 2837	1 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶))} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  {cab 2711  ∀wral 3048  ∃wrex 3057  Vcvv 3437   ∪ cun 3896   ⊆ wss 3898  ∅c0 4282  {copab 5157  ωcom 7804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676  ax-inf2 9540

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-tr 5203  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-om 7805

This theorem is referenced by:  satf0suc  35443  sat1el2xp  35446