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Theorem kmlem9 10152
Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
Hypothesis
Ref Expression
kmlem9.1 𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}
Assertion
Ref Expression
kmlem9 𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)
Distinct variable groups:   𝑥,𝑧,𝑤,𝑢,𝑡   𝑧,𝐴,𝑤
Allowed substitution hints:   𝐴(𝑥,𝑢,𝑡)

Proof of Theorem kmlem9
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 3478 . . . 4 𝑧 ∈ V
2 eqeq1 2736 . . . . 5 (𝑢 = 𝑧 → (𝑢 = (𝑡 (𝑥 ∖ {𝑡})) ↔ 𝑧 = (𝑡 (𝑥 ∖ {𝑡}))))
32rexbidv 3178 . . . 4 (𝑢 = 𝑧 → (∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡})) ↔ ∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡}))))
4 kmlem9.1 . . . 4 𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}
51, 3, 4elab2 3672 . . 3 (𝑧𝐴 ↔ ∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡})))
6 vex 3478 . . . . 5 𝑤 ∈ V
7 eqeq1 2736 . . . . . 6 (𝑢 = 𝑤 → (𝑢 = (𝑡 (𝑥 ∖ {𝑡})) ↔ 𝑤 = (𝑡 (𝑥 ∖ {𝑡}))))
87rexbidv 3178 . . . . 5 (𝑢 = 𝑤 → (∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡})) ↔ ∃𝑡𝑥 𝑤 = (𝑡 (𝑥 ∖ {𝑡}))))
96, 8, 4elab2 3672 . . . 4 (𝑤𝐴 ↔ ∃𝑡𝑥 𝑤 = (𝑡 (𝑥 ∖ {𝑡})))
10 difeq1 4115 . . . . . . 7 (𝑡 = → (𝑡 (𝑥 ∖ {𝑡})) = ( (𝑥 ∖ {𝑡})))
11 sneq 4638 . . . . . . . . . 10 (𝑡 = → {𝑡} = {})
1211difeq2d 4122 . . . . . . . . 9 (𝑡 = → (𝑥 ∖ {𝑡}) = (𝑥 ∖ {}))
1312unieqd 4922 . . . . . . . 8 (𝑡 = (𝑥 ∖ {𝑡}) = (𝑥 ∖ {}))
1413difeq2d 4122 . . . . . . 7 (𝑡 = → ( (𝑥 ∖ {𝑡})) = ( (𝑥 ∖ {})))
1510, 14eqtrd 2772 . . . . . 6 (𝑡 = → (𝑡 (𝑥 ∖ {𝑡})) = ( (𝑥 ∖ {})))
1615eqeq2d 2743 . . . . 5 (𝑡 = → (𝑤 = (𝑡 (𝑥 ∖ {𝑡})) ↔ 𝑤 = ( (𝑥 ∖ {}))))
1716cbvrexvw 3235 . . . 4 (∃𝑡𝑥 𝑤 = (𝑡 (𝑥 ∖ {𝑡})) ↔ ∃𝑥 𝑤 = ( (𝑥 ∖ {})))
189, 17bitri 274 . . 3 (𝑤𝐴 ↔ ∃𝑥 𝑤 = ( (𝑥 ∖ {})))
19 reeanv 3226 . . . 4 (∃𝑡𝑥𝑥 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) ↔ (∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ ∃𝑥 𝑤 = ( (𝑥 ∖ {}))))
20 eqeq12 2749 . . . . . . . . . 10 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧 = 𝑤 ↔ (𝑡 (𝑥 ∖ {𝑡})) = ( (𝑥 ∖ {}))))
2115, 20imbitrrid 245 . . . . . . . . 9 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑡 = 𝑧 = 𝑤))
2221necon3d 2961 . . . . . . . 8 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤𝑡))
23 kmlem5 10148 . . . . . . . . . 10 ((𝑥𝑡) → ((𝑡 (𝑥 ∖ {𝑡})) ∩ ( (𝑥 ∖ {}))) = ∅)
24 ineq12 4207 . . . . . . . . . . 11 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤) = ((𝑡 (𝑥 ∖ {𝑡})) ∩ ( (𝑥 ∖ {}))))
2524eqeq1d 2734 . . . . . . . . . 10 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → ((𝑧𝑤) = ∅ ↔ ((𝑡 (𝑥 ∖ {𝑡})) ∩ ( (𝑥 ∖ {}))) = ∅))
2623, 25imbitrrid 245 . . . . . . . . 9 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → ((𝑥𝑡) → (𝑧𝑤) = ∅))
2726expd 416 . . . . . . . 8 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑥 → (𝑡 → (𝑧𝑤) = ∅)))
2822, 27syl5d 73 . . . . . . 7 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑥 → (𝑧𝑤 → (𝑧𝑤) = ∅)))
2928com12 32 . . . . . 6 (𝑥 → ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤 → (𝑧𝑤) = ∅)))
3029adantl 482 . . . . 5 ((𝑡𝑥𝑥) → ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤 → (𝑧𝑤) = ∅)))
3130rexlimivv 3199 . . . 4 (∃𝑡𝑥𝑥 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤 → (𝑧𝑤) = ∅))
3219, 31sylbir 234 . . 3 ((∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ ∃𝑥 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤 → (𝑧𝑤) = ∅))
335, 18, 32syl2anb 598 . 2 ((𝑧𝐴𝑤𝐴) → (𝑧𝑤 → (𝑧𝑤) = ∅))
3433rgen2 3197 1 𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {cab 2709  wne 2940  wral 3061  wrex 3070  cdif 3945  cin 3947  c0 4322  {csn 4628   cuni 4908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-uni 4909
This theorem is referenced by:  kmlem10  10153
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