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Theorem isf32lem3 10350
Description: Lemma for isfin3-2 10362. Being a chain, difference sets are disjoint (one case). (Contributed by Stefan O'Rear, 5-Nov-2014.)
Hypotheses
Ref Expression
isf32lem.a (πœ‘ β†’ 𝐹:Ο‰βŸΆπ’« 𝐺)
isf32lem.b (πœ‘ β†’ βˆ€π‘₯ ∈ Ο‰ (πΉβ€˜suc π‘₯) βŠ† (πΉβ€˜π‘₯))
isf32lem.c (πœ‘ β†’ Β¬ ∩ ran 𝐹 ∈ ran 𝐹)
Assertion
Ref Expression
isf32lem3 (((𝐴 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ (𝐡 ∈ 𝐴 ∧ πœ‘)) β†’ (((πΉβ€˜π΄) βˆ– (πΉβ€˜suc 𝐴)) ∩ ((πΉβ€˜π΅) βˆ– (πΉβ€˜suc 𝐡))) = βˆ…)
Distinct variable groups:   π‘₯,𝐡   πœ‘,π‘₯   π‘₯,𝐴   π‘₯,𝐹
Allowed substitution hint:   𝐺(π‘₯)

Proof of Theorem isf32lem3
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 eldifi 4127 . . . 4 (π‘Ž ∈ ((πΉβ€˜π΄) βˆ– (πΉβ€˜suc 𝐴)) β†’ π‘Ž ∈ (πΉβ€˜π΄))
2 simpll 766 . . . . . 6 (((𝐴 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ (𝐡 ∈ 𝐴 ∧ πœ‘)) β†’ 𝐴 ∈ Ο‰)
3 peano2 7881 . . . . . . 7 (𝐡 ∈ Ο‰ β†’ suc 𝐡 ∈ Ο‰)
43ad2antlr 726 . . . . . 6 (((𝐴 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ (𝐡 ∈ 𝐴 ∧ πœ‘)) β†’ suc 𝐡 ∈ Ο‰)
5 nnord 7863 . . . . . . . 8 (𝐴 ∈ Ο‰ β†’ Ord 𝐴)
65ad2antrr 725 . . . . . . 7 (((𝐴 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ (𝐡 ∈ 𝐴 ∧ πœ‘)) β†’ Ord 𝐴)
7 simprl 770 . . . . . . 7 (((𝐴 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ (𝐡 ∈ 𝐴 ∧ πœ‘)) β†’ 𝐡 ∈ 𝐴)
8 ordsucss 7806 . . . . . . 7 (Ord 𝐴 β†’ (𝐡 ∈ 𝐴 β†’ suc 𝐡 βŠ† 𝐴))
96, 7, 8sylc 65 . . . . . 6 (((𝐴 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ (𝐡 ∈ 𝐴 ∧ πœ‘)) β†’ suc 𝐡 βŠ† 𝐴)
10 simprr 772 . . . . . 6 (((𝐴 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ (𝐡 ∈ 𝐴 ∧ πœ‘)) β†’ πœ‘)
11 isf32lem.a . . . . . . 7 (πœ‘ β†’ 𝐹:Ο‰βŸΆπ’« 𝐺)
12 isf32lem.b . . . . . . 7 (πœ‘ β†’ βˆ€π‘₯ ∈ Ο‰ (πΉβ€˜suc π‘₯) βŠ† (πΉβ€˜π‘₯))
13 isf32lem.c . . . . . . 7 (πœ‘ β†’ Β¬ ∩ ran 𝐹 ∈ ran 𝐹)
1411, 12, 13isf32lem1 10348 . . . . . 6 (((𝐴 ∈ Ο‰ ∧ suc 𝐡 ∈ Ο‰) ∧ (suc 𝐡 βŠ† 𝐴 ∧ πœ‘)) β†’ (πΉβ€˜π΄) βŠ† (πΉβ€˜suc 𝐡))
152, 4, 9, 10, 14syl22anc 838 . . . . 5 (((𝐴 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ (𝐡 ∈ 𝐴 ∧ πœ‘)) β†’ (πΉβ€˜π΄) βŠ† (πΉβ€˜suc 𝐡))
1615sseld 3982 . . . 4 (((𝐴 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ (𝐡 ∈ 𝐴 ∧ πœ‘)) β†’ (π‘Ž ∈ (πΉβ€˜π΄) β†’ π‘Ž ∈ (πΉβ€˜suc 𝐡)))
17 elndif 4129 . . . 4 (π‘Ž ∈ (πΉβ€˜suc 𝐡) β†’ Β¬ π‘Ž ∈ ((πΉβ€˜π΅) βˆ– (πΉβ€˜suc 𝐡)))
181, 16, 17syl56 36 . . 3 (((𝐴 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ (𝐡 ∈ 𝐴 ∧ πœ‘)) β†’ (π‘Ž ∈ ((πΉβ€˜π΄) βˆ– (πΉβ€˜suc 𝐴)) β†’ Β¬ π‘Ž ∈ ((πΉβ€˜π΅) βˆ– (πΉβ€˜suc 𝐡))))
1918ralrimiv 3146 . 2 (((𝐴 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ (𝐡 ∈ 𝐴 ∧ πœ‘)) β†’ βˆ€π‘Ž ∈ ((πΉβ€˜π΄) βˆ– (πΉβ€˜suc 𝐴)) Β¬ π‘Ž ∈ ((πΉβ€˜π΅) βˆ– (πΉβ€˜suc 𝐡)))
20 disj 4448 . 2 ((((πΉβ€˜π΄) βˆ– (πΉβ€˜suc 𝐴)) ∩ ((πΉβ€˜π΅) βˆ– (πΉβ€˜suc 𝐡))) = βˆ… ↔ βˆ€π‘Ž ∈ ((πΉβ€˜π΄) βˆ– (πΉβ€˜suc 𝐴)) Β¬ π‘Ž ∈ ((πΉβ€˜π΅) βˆ– (πΉβ€˜suc 𝐡)))
2119, 20sylibr 233 1 (((𝐴 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ (𝐡 ∈ 𝐴 ∧ πœ‘)) β†’ (((πΉβ€˜π΄) βˆ– (πΉβ€˜suc 𝐴)) ∩ ((πΉβ€˜π΅) βˆ– (πΉβ€˜suc 𝐡))) = βˆ…)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  βˆ© cint 4951  ran crn 5678  Ord word 6364  suc csuc 6367  βŸΆwf 6540  β€˜cfv 6544  Ο‰com 7855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fv 6552  df-om 7856
This theorem is referenced by:  isf32lem4  10351
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