Theorem satf0suclem 34664

Description: Lemma for satf0suc 34665, sat1el2xp 34668 and fmlasuc0 34673. (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 7881	. . . . . 6 ⊢ ∅ ∈ ω
2		eleq1 2819	. . . . . 6 ⊢ (𝑦 = ∅ → (𝑦 ∈ ω ↔ ∅ ∈ ω))
3	1, 2	mpbiri 257	. . . . 5 ⊢ (𝑦 = ∅ → 𝑦 ∈ ω)
4	3	adantr 479	. . . 4 ⊢ ((𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)) → 𝑦 ∈ ω)
5	4	pm4.71ri 559	. . 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 5214	. 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 9640	. . . . 5 ⊢ ω ∈ V
8	7	a1i 11	. . . 4 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ω ∈ V)
9		simp1 1134	. . . . . 6 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑋 ∈ 𝑈)
10		unab 4297	. . . . . . . 8 ⊢ ({𝑥 ∣ ∃𝑣 ∈ 𝑌 𝑥 = 𝐵} ∪ {𝑥 ∣ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶}) = {𝑥 ∣ (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)}
11		abrexexg 7949	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝑉 → {𝑥 ∣ ∃𝑣 ∈ 𝑌 𝑥 = 𝐵} ∈ V)
12	11	3ad2ant2 1132	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑥 ∣ ∃𝑣 ∈ 𝑌 𝑥 = 𝐵} ∈ V)
13		abrexexg 7949	. . . . . . . . . 10 ⊢ (𝑍 ∈ 𝑊 → {𝑥 ∣ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶} ∈ V)
14	13	3ad2ant3 1133	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑥 ∣ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶} ∈ V)
15		unexg 7738	. . . . . . . . 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 582	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ({𝑥 ∣ ∃𝑣 ∈ 𝑌 𝑥 = 𝐵} ∪ {𝑥 ∣ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶}) ∈ V)
17	10, 16	eqeltrrid 2836	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑥 ∣ (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)} ∈ V)
18	17	ralrimivw 3148	. . . . . 6 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ∀𝑢 ∈ 𝑋 {𝑥 ∣ (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)} ∈ V)
19		abrexex2g 7953	. . . . . 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 582	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑥 ∣ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶)} ∈ V)
21	20	adantr 479	. . . 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 7955	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ ω ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶))} ∈ V)
23		simpr 483	. . . . . 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 615	. . . . 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 5549	. . . 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 5323	. 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 2835	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 394   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  {cab 2707  ∀wral 3059  ∃wrex 3068  Vcvv 3472   ∪ cun 3945   ⊆ wss 3947  ∅c0 4321  {copab 5209  ωcom 7857

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-om 7858

This theorem is referenced by:  satf0suc  34665  sat1el2xp  34668