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Theorem satf0suclem 35360
Description: Lemma for satf0suc 35361, sat1el2xp 35364 and fmlasuc0 35369. (Contributed by AV, 19-Sep-2023.)
Assertion
Ref Expression
satf0suclem ((𝑋𝑈𝑌𝑉𝑍𝑊) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 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
StepHypRef Expression
1 peano1 7911 . . . . . 6 ∅ ∈ ω
2 eleq1 2827 . . . . . 6 (𝑦 = ∅ → (𝑦 ∈ ω ↔ ∅ ∈ ω))
31, 2mpbiri 258 . . . . 5 (𝑦 = ∅ → 𝑦 ∈ ω)
43adantr 480 . . . 4 ((𝑦 = ∅ ∧ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶)) → 𝑦 ∈ ω)
54pm4.71ri 560 . . 3 ((𝑦 = ∅ ∧ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶)) ↔ (𝑦 ∈ ω ∧ (𝑦 = ∅ ∧ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶))))
65opabbii 5215 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶))} = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ ω ∧ (𝑦 = ∅ ∧ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶)))}
7 omex 9681 . . . . 5 ω ∈ V
87a1i 11 . . . 4 ((𝑋𝑈𝑌𝑉𝑍𝑊) → ω ∈ V)
9 simp1 1135 . . . . . 6 ((𝑋𝑈𝑌𝑉𝑍𝑊) → 𝑋𝑈)
10 unab 4314 . . . . . . . 8 ({𝑥 ∣ ∃𝑣𝑌 𝑥 = 𝐵} ∪ {𝑥 ∣ ∃𝑤𝑍 𝑥 = 𝐶}) = {𝑥 ∣ (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶)}
11 abrexexg 7984 . . . . . . . . . 10 (𝑌𝑉 → {𝑥 ∣ ∃𝑣𝑌 𝑥 = 𝐵} ∈ V)
12113ad2ant2 1133 . . . . . . . . 9 ((𝑋𝑈𝑌𝑉𝑍𝑊) → {𝑥 ∣ ∃𝑣𝑌 𝑥 = 𝐵} ∈ V)
13 abrexexg 7984 . . . . . . . . . 10 (𝑍𝑊 → {𝑥 ∣ ∃𝑤𝑍 𝑥 = 𝐶} ∈ V)
14133ad2ant3 1134 . . . . . . . . 9 ((𝑋𝑈𝑌𝑉𝑍𝑊) → {𝑥 ∣ ∃𝑤𝑍 𝑥 = 𝐶} ∈ V)
15 unexg 7762 . . . . . . . . 9 (({𝑥 ∣ ∃𝑣𝑌 𝑥 = 𝐵} ∈ V ∧ {𝑥 ∣ ∃𝑤𝑍 𝑥 = 𝐶} ∈ V) → ({𝑥 ∣ ∃𝑣𝑌 𝑥 = 𝐵} ∪ {𝑥 ∣ ∃𝑤𝑍 𝑥 = 𝐶}) ∈ V)
1612, 14, 15syl2anc 584 . . . . . . . 8 ((𝑋𝑈𝑌𝑉𝑍𝑊) → ({𝑥 ∣ ∃𝑣𝑌 𝑥 = 𝐵} ∪ {𝑥 ∣ ∃𝑤𝑍 𝑥 = 𝐶}) ∈ V)
1710, 16eqeltrrid 2844 . . . . . . 7 ((𝑋𝑈𝑌𝑉𝑍𝑊) → {𝑥 ∣ (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶)} ∈ V)
1817ralrimivw 3148 . . . . . 6 ((𝑋𝑈𝑌𝑉𝑍𝑊) → ∀𝑢𝑋 {𝑥 ∣ (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶)} ∈ V)
19 abrexex2g 7988 . . . . . 6 ((𝑋𝑈 ∧ ∀𝑢𝑋 {𝑥 ∣ (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶)} ∈ V) → {𝑥 ∣ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶)} ∈ V)
209, 18, 19syl2anc 584 . . . . 5 ((𝑋𝑈𝑌𝑉𝑍𝑊) → {𝑥 ∣ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶)} ∈ V)
2120adantr 480 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑦 ∈ ω) → {𝑥 ∣ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶)} ∈ V)
228, 21opabex3rd 7990 . . 3 ((𝑋𝑈𝑌𝑉𝑍𝑊) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ ω ∧ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶))} ∈ V)
23 simpr 484 . . . . . 6 ((𝑦 = ∅ ∧ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶)) → ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶))
2423anim2i 617 . . . . 5 ((𝑦 ∈ ω ∧ (𝑦 = ∅ ∧ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶))) → (𝑦 ∈ ω ∧ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶)))
2524ssopab2i 5560 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ ω ∧ (𝑦 = ∅ ∧ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶)))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ ω ∧ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶))}
2625a1i 11 . . 3 ((𝑋𝑈𝑌𝑉𝑍𝑊) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ ω ∧ (𝑦 = ∅ ∧ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶)))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ ω ∧ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶))})
2722, 26ssexd 5330 . 2 ((𝑋𝑈𝑌𝑉𝑍𝑊) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ ω ∧ (𝑦 = ∅ ∧ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶)))} ∈ V)
286, 27eqeltrid 2843 1 ((𝑋𝑈𝑌𝑉𝑍𝑊) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶))} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  cun 3961  wss 3963  c0 4339  {copab 5210  ωcom 7887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-om 7888
This theorem is referenced by:  satf0suc  35361  sat1el2xp  35364
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