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Theorem kmlem9 9631
 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 3413 . . . 4 𝑧 ∈ V
2 eqeq1 2762 . . . . 5 (𝑢 = 𝑧 → (𝑢 = (𝑡 (𝑥 ∖ {𝑡})) ↔ 𝑧 = (𝑡 (𝑥 ∖ {𝑡}))))
32rexbidv 3221 . . . 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 3593 . . 3 (𝑧𝐴 ↔ ∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡})))
6 vex 3413 . . . . 5 𝑤 ∈ V
7 eqeq1 2762 . . . . . 6 (𝑢 = 𝑤 → (𝑢 = (𝑡 (𝑥 ∖ {𝑡})) ↔ 𝑤 = (𝑡 (𝑥 ∖ {𝑡}))))
87rexbidv 3221 . . . . 5 (𝑢 = 𝑤 → (∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡})) ↔ ∃𝑡𝑥 𝑤 = (𝑡 (𝑥 ∖ {𝑡}))))
96, 8, 4elab2 3593 . . . 4 (𝑤𝐴 ↔ ∃𝑡𝑥 𝑤 = (𝑡 (𝑥 ∖ {𝑡})))
10 difeq1 4023 . . . . . . 7 (𝑡 = → (𝑡 (𝑥 ∖ {𝑡})) = ( (𝑥 ∖ {𝑡})))
11 sneq 4535 . . . . . . . . . 10 (𝑡 = → {𝑡} = {})
1211difeq2d 4030 . . . . . . . . 9 (𝑡 = → (𝑥 ∖ {𝑡}) = (𝑥 ∖ {}))
1312unieqd 4815 . . . . . . . 8 (𝑡 = (𝑥 ∖ {𝑡}) = (𝑥 ∖ {}))
1413difeq2d 4030 . . . . . . 7 (𝑡 = → ( (𝑥 ∖ {𝑡})) = ( (𝑥 ∖ {})))
1510, 14eqtrd 2793 . . . . . 6 (𝑡 = → (𝑡 (𝑥 ∖ {𝑡})) = ( (𝑥 ∖ {})))
1615eqeq2d 2769 . . . . 5 (𝑡 = → (𝑤 = (𝑡 (𝑥 ∖ {𝑡})) ↔ 𝑤 = ( (𝑥 ∖ {}))))
1716cbvrexvw 3362 . . . 4 (∃𝑡𝑥 𝑤 = (𝑡 (𝑥 ∖ {𝑡})) ↔ ∃𝑥 𝑤 = ( (𝑥 ∖ {})))
189, 17bitri 278 . . 3 (𝑤𝐴 ↔ ∃𝑥 𝑤 = ( (𝑥 ∖ {})))
19 reeanv 3285 . . . 4 (∃𝑡𝑥𝑥 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) ↔ (∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ ∃𝑥 𝑤 = ( (𝑥 ∖ {}))))
20 eqeq12 2772 . . . . . . . . . 10 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧 = 𝑤 ↔ (𝑡 (𝑥 ∖ {𝑡})) = ( (𝑥 ∖ {}))))
2115, 20syl5ibr 249 . . . . . . . . 9 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑡 = 𝑧 = 𝑤))
2221necon3d 2972 . . . . . . . 8 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤𝑡))
23 kmlem5 9627 . . . . . . . . . 10 ((𝑥𝑡) → ((𝑡 (𝑥 ∖ {𝑡})) ∩ ( (𝑥 ∖ {}))) = ∅)
24 ineq12 4114 . . . . . . . . . . 11 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤) = ((𝑡 (𝑥 ∖ {𝑡})) ∩ ( (𝑥 ∖ {}))))
2524eqeq1d 2760 . . . . . . . . . 10 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → ((𝑧𝑤) = ∅ ↔ ((𝑡 (𝑥 ∖ {𝑡})) ∩ ( (𝑥 ∖ {}))) = ∅))
2623, 25syl5ibr 249 . . . . . . . . 9 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → ((𝑥𝑡) → (𝑧𝑤) = ∅))
2726expd 419 . . . . . . . 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 485 . . . . 5 ((𝑡𝑥𝑥) → ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤 → (𝑧𝑤) = ∅)))
3130rexlimivv 3216 . . . 4 (∃𝑡𝑥𝑥 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤 → (𝑧𝑤) = ∅))
3219, 31sylbir 238 . . 3 ((∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ ∃𝑥 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤 → (𝑧𝑤) = ∅))
335, 18, 32syl2anb 600 . 2 ((𝑧𝐴𝑤𝐴) → (𝑧𝑤 → (𝑧𝑤) = ∅))
3433rgen2 3132 1 𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735   ≠ wne 2951  ∀wral 3070  ∃wrex 3071   ∖ cdif 3857   ∩ cin 3859  ∅c0 4227  {csn 4525  ∪ cuni 4801 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-in 3867  df-ss 3877  df-nul 4228  df-sn 4526  df-uni 4802 This theorem is referenced by:  kmlem10  9632
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