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Theorem kmlem9 8863
 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 3176 . . . 4 𝑧 ∈ V
2 eqeq1 2614 . . . . 5 (𝑢 = 𝑧 → (𝑢 = (𝑡 (𝑥 ∖ {𝑡})) ↔ 𝑧 = (𝑡 (𝑥 ∖ {𝑡}))))
32rexbidv 3034 . . . 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 3323 . . 3 (𝑧𝐴 ↔ ∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡})))
6 vex 3176 . . . . 5 𝑤 ∈ V
7 eqeq1 2614 . . . . . 6 (𝑢 = 𝑤 → (𝑢 = (𝑡 (𝑥 ∖ {𝑡})) ↔ 𝑤 = (𝑡 (𝑥 ∖ {𝑡}))))
87rexbidv 3034 . . . . 5 (𝑢 = 𝑤 → (∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡})) ↔ ∃𝑡𝑥 𝑤 = (𝑡 (𝑥 ∖ {𝑡}))))
96, 8, 4elab2 3323 . . . 4 (𝑤𝐴 ↔ ∃𝑡𝑥 𝑤 = (𝑡 (𝑥 ∖ {𝑡})))
10 difeq1 3683 . . . . . . 7 (𝑡 = → (𝑡 (𝑥 ∖ {𝑡})) = ( (𝑥 ∖ {𝑡})))
11 sneq 4135 . . . . . . . . . 10 (𝑡 = → {𝑡} = {})
1211difeq2d 3690 . . . . . . . . 9 (𝑡 = → (𝑥 ∖ {𝑡}) = (𝑥 ∖ {}))
1312unieqd 4382 . . . . . . . 8 (𝑡 = (𝑥 ∖ {𝑡}) = (𝑥 ∖ {}))
1413difeq2d 3690 . . . . . . 7 (𝑡 = → ( (𝑥 ∖ {𝑡})) = ( (𝑥 ∖ {})))
1510, 14eqtrd 2644 . . . . . 6 (𝑡 = → (𝑡 (𝑥 ∖ {𝑡})) = ( (𝑥 ∖ {})))
1615eqeq2d 2620 . . . . 5 (𝑡 = → (𝑤 = (𝑡 (𝑥 ∖ {𝑡})) ↔ 𝑤 = ( (𝑥 ∖ {}))))
1716cbvrexv 3148 . . . 4 (∃𝑡𝑥 𝑤 = (𝑡 (𝑥 ∖ {𝑡})) ↔ ∃𝑥 𝑤 = ( (𝑥 ∖ {})))
189, 17bitri 263 . . 3 (𝑤𝐴 ↔ ∃𝑥 𝑤 = ( (𝑥 ∖ {})))
19 reeanv 3086 . . . 4 (∃𝑡𝑥𝑥 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) ↔ (∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ ∃𝑥 𝑤 = ( (𝑥 ∖ {}))))
20 eqeq12 2623 . . . . . . . . . 10 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧 = 𝑤 ↔ (𝑡 (𝑥 ∖ {𝑡})) = ( (𝑥 ∖ {}))))
2115, 20syl5ibr 235 . . . . . . . . 9 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑡 = 𝑧 = 𝑤))
2221necon3d 2803 . . . . . . . 8 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤𝑡))
23 kmlem5 8859 . . . . . . . . . 10 ((𝑥𝑡) → ((𝑡 (𝑥 ∖ {𝑡})) ∩ ( (𝑥 ∖ {}))) = ∅)
24 ineq12 3771 . . . . . . . . . . 11 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤) = ((𝑡 (𝑥 ∖ {𝑡})) ∩ ( (𝑥 ∖ {}))))
2524eqeq1d 2612 . . . . . . . . . 10 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → ((𝑧𝑤) = ∅ ↔ ((𝑡 (𝑥 ∖ {𝑡})) ∩ ( (𝑥 ∖ {}))) = ∅))
2623, 25syl5ibr 235 . . . . . . . . 9 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → ((𝑥𝑡) → (𝑧𝑤) = ∅))
2726expd 451 . . . . . . . 8 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑥 → (𝑡 → (𝑧𝑤) = ∅)))
2822, 27syl5d 71 . . . . . . 7 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑥 → (𝑧𝑤 → (𝑧𝑤) = ∅)))
2928com12 32 . . . . . 6 (𝑥 → ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤 → (𝑧𝑤) = ∅)))
3029adantl 481 . . . . 5 ((𝑡𝑥𝑥) → ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤 → (𝑧𝑤) = ∅)))
3130rexlimivv 3018 . . . 4 (∃𝑡𝑥𝑥 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤 → (𝑧𝑤) = ∅))
3219, 31sylbir 224 . . 3 ((∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ ∃𝑥 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤 → (𝑧𝑤) = ∅))
335, 18, 32syl2anb 495 . 2 ((𝑧𝐴𝑤𝐴) → (𝑧𝑤 → (𝑧𝑤) = ∅))
3433rgen2a 2960 1 𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537   ∩ cin 3539  ∅c0 3874  {csn 4125  ∪ cuni 4372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-uni 4373 This theorem is referenced by:  kmlem10  8864
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