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Theorem kmlem9 8927
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 3189 . . . 4 𝑧 ∈ V
2 eqeq1 2625 . . . . 5 (𝑢 = 𝑧 → (𝑢 = (𝑡 (𝑥 ∖ {𝑡})) ↔ 𝑧 = (𝑡 (𝑥 ∖ {𝑡}))))
32rexbidv 3045 . . . 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 3338 . . 3 (𝑧𝐴 ↔ ∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡})))
6 vex 3189 . . . . 5 𝑤 ∈ V
7 eqeq1 2625 . . . . . 6 (𝑢 = 𝑤 → (𝑢 = (𝑡 (𝑥 ∖ {𝑡})) ↔ 𝑤 = (𝑡 (𝑥 ∖ {𝑡}))))
87rexbidv 3045 . . . . 5 (𝑢 = 𝑤 → (∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡})) ↔ ∃𝑡𝑥 𝑤 = (𝑡 (𝑥 ∖ {𝑡}))))
96, 8, 4elab2 3338 . . . 4 (𝑤𝐴 ↔ ∃𝑡𝑥 𝑤 = (𝑡 (𝑥 ∖ {𝑡})))
10 difeq1 3701 . . . . . . 7 (𝑡 = → (𝑡 (𝑥 ∖ {𝑡})) = ( (𝑥 ∖ {𝑡})))
11 sneq 4160 . . . . . . . . . 10 (𝑡 = → {𝑡} = {})
1211difeq2d 3708 . . . . . . . . 9 (𝑡 = → (𝑥 ∖ {𝑡}) = (𝑥 ∖ {}))
1312unieqd 4414 . . . . . . . 8 (𝑡 = (𝑥 ∖ {𝑡}) = (𝑥 ∖ {}))
1413difeq2d 3708 . . . . . . 7 (𝑡 = → ( (𝑥 ∖ {𝑡})) = ( (𝑥 ∖ {})))
1510, 14eqtrd 2655 . . . . . 6 (𝑡 = → (𝑡 (𝑥 ∖ {𝑡})) = ( (𝑥 ∖ {})))
1615eqeq2d 2631 . . . . 5 (𝑡 = → (𝑤 = (𝑡 (𝑥 ∖ {𝑡})) ↔ 𝑤 = ( (𝑥 ∖ {}))))
1716cbvrexv 3160 . . . 4 (∃𝑡𝑥 𝑤 = (𝑡 (𝑥 ∖ {𝑡})) ↔ ∃𝑥 𝑤 = ( (𝑥 ∖ {})))
189, 17bitri 264 . . 3 (𝑤𝐴 ↔ ∃𝑥 𝑤 = ( (𝑥 ∖ {})))
19 reeanv 3097 . . . 4 (∃𝑡𝑥𝑥 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) ↔ (∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ ∃𝑥 𝑤 = ( (𝑥 ∖ {}))))
20 eqeq12 2634 . . . . . . . . . 10 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧 = 𝑤 ↔ (𝑡 (𝑥 ∖ {𝑡})) = ( (𝑥 ∖ {}))))
2115, 20syl5ibr 236 . . . . . . . . 9 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑡 = 𝑧 = 𝑤))
2221necon3d 2811 . . . . . . . 8 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤𝑡))
23 kmlem5 8923 . . . . . . . . . 10 ((𝑥𝑡) → ((𝑡 (𝑥 ∖ {𝑡})) ∩ ( (𝑥 ∖ {}))) = ∅)
24 ineq12 3789 . . . . . . . . . . 11 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤) = ((𝑡 (𝑥 ∖ {𝑡})) ∩ ( (𝑥 ∖ {}))))
2524eqeq1d 2623 . . . . . . . . . 10 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → ((𝑧𝑤) = ∅ ↔ ((𝑡 (𝑥 ∖ {𝑡})) ∩ ( (𝑥 ∖ {}))) = ∅))
2623, 25syl5ibr 236 . . . . . . . . 9 ((𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → ((𝑥𝑡) → (𝑧𝑤) = ∅))
2726expd 452 . . . . . . . 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 3029 . . . 4 (∃𝑡𝑥𝑥 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤 → (𝑧𝑤) = ∅))
3219, 31sylbir 225 . . 3 ((∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡})) ∧ ∃𝑥 𝑤 = ( (𝑥 ∖ {}))) → (𝑧𝑤 → (𝑧𝑤) = ∅))
335, 18, 32syl2anb 496 . 2 ((𝑧𝐴𝑤𝐴) → (𝑧𝑤 → (𝑧𝑤) = ∅))
3433rgen2a 2971 1 𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {cab 2607  wne 2790  wral 2907  wrex 2908  cdif 3553  cin 3555  c0 3893  {csn 4150   cuni 4404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3559  df-in 3563  df-ss 3570  df-nul 3894  df-sn 4151  df-uni 4405
This theorem is referenced by:  kmlem10  8928
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