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Theorem fin1a2lem9 10403
Description: Lemma for fin1a2 10410. In a chain of finite sets, initial segments are finite. (Contributed by Stefan O'Rear, 8-Nov-2014.)
Assertion
Ref Expression
fin1a2lem9 (( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) β†’ {𝑏 ∈ 𝑋 ∣ 𝑏 β‰Ό 𝐴} ∈ Fin)
Distinct variable groups:   𝐴,𝑏   𝑋,𝑏

Proof of Theorem fin1a2lem9
Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onfin2 9231 . . . . 5 Ο‰ = (On ∩ Fin)
2 inss2 4230 . . . . 5 (On ∩ Fin) βŠ† Fin
31, 2eqsstri 4017 . . . 4 Ο‰ βŠ† Fin
4 peano2 7881 . . . 4 (𝐴 ∈ Ο‰ β†’ suc 𝐴 ∈ Ο‰)
53, 4sselid 3981 . . 3 (𝐴 ∈ Ο‰ β†’ suc 𝐴 ∈ Fin)
653ad2ant3 1136 . 2 (( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) β†’ suc 𝐴 ∈ Fin)
743ad2ant3 1136 . . 3 (( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) β†’ suc 𝐴 ∈ Ο‰)
8 breq1 5152 . . . . . 6 (𝑏 = 𝑐 β†’ (𝑏 β‰Ό 𝐴 ↔ 𝑐 β‰Ό 𝐴))
98elrab 3684 . . . . 5 (𝑐 ∈ {𝑏 ∈ 𝑋 ∣ 𝑏 β‰Ό 𝐴} ↔ (𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴))
10 simprr 772 . . . . . . . 8 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴)) β†’ 𝑐 β‰Ό 𝐴)
11 simpl2 1193 . . . . . . . . . . 11 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴)) β†’ 𝑋 βŠ† Fin)
12 simprl 770 . . . . . . . . . . 11 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴)) β†’ 𝑐 ∈ 𝑋)
1311, 12sseldd 3984 . . . . . . . . . 10 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴)) β†’ 𝑐 ∈ Fin)
14 finnum 9943 . . . . . . . . . 10 (𝑐 ∈ Fin β†’ 𝑐 ∈ dom card)
1513, 14syl 17 . . . . . . . . 9 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴)) β†’ 𝑐 ∈ dom card)
16 simpl3 1194 . . . . . . . . . . 11 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴)) β†’ 𝐴 ∈ Ο‰)
173, 16sselid 3981 . . . . . . . . . 10 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴)) β†’ 𝐴 ∈ Fin)
18 finnum 9943 . . . . . . . . . 10 (𝐴 ∈ Fin β†’ 𝐴 ∈ dom card)
1917, 18syl 17 . . . . . . . . 9 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴)) β†’ 𝐴 ∈ dom card)
20 carddom2 9972 . . . . . . . . 9 ((𝑐 ∈ dom card ∧ 𝐴 ∈ dom card) β†’ ((cardβ€˜π‘) βŠ† (cardβ€˜π΄) ↔ 𝑐 β‰Ό 𝐴))
2115, 19, 20syl2anc 585 . . . . . . . 8 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴)) β†’ ((cardβ€˜π‘) βŠ† (cardβ€˜π΄) ↔ 𝑐 β‰Ό 𝐴))
2210, 21mpbird 257 . . . . . . 7 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴)) β†’ (cardβ€˜π‘) βŠ† (cardβ€˜π΄))
2322ex 414 . . . . . 6 (( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) β†’ ((𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴) β†’ (cardβ€˜π‘) βŠ† (cardβ€˜π΄)))
24 cardnn 9958 . . . . . . . . 9 (𝐴 ∈ Ο‰ β†’ (cardβ€˜π΄) = 𝐴)
2524sseq2d 4015 . . . . . . . 8 (𝐴 ∈ Ο‰ β†’ ((cardβ€˜π‘) βŠ† (cardβ€˜π΄) ↔ (cardβ€˜π‘) βŠ† 𝐴))
26 cardon 9939 . . . . . . . . 9 (cardβ€˜π‘) ∈ On
27 nnon 7861 . . . . . . . . 9 (𝐴 ∈ Ο‰ β†’ 𝐴 ∈ On)
28 onsssuc 6455 . . . . . . . . 9 (((cardβ€˜π‘) ∈ On ∧ 𝐴 ∈ On) β†’ ((cardβ€˜π‘) βŠ† 𝐴 ↔ (cardβ€˜π‘) ∈ suc 𝐴))
2926, 27, 28sylancr 588 . . . . . . . 8 (𝐴 ∈ Ο‰ β†’ ((cardβ€˜π‘) βŠ† 𝐴 ↔ (cardβ€˜π‘) ∈ suc 𝐴))
3025, 29bitrd 279 . . . . . . 7 (𝐴 ∈ Ο‰ β†’ ((cardβ€˜π‘) βŠ† (cardβ€˜π΄) ↔ (cardβ€˜π‘) ∈ suc 𝐴))
31303ad2ant3 1136 . . . . . 6 (( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) β†’ ((cardβ€˜π‘) βŠ† (cardβ€˜π΄) ↔ (cardβ€˜π‘) ∈ suc 𝐴))
3223, 31sylibd 238 . . . . 5 (( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) β†’ ((𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴) β†’ (cardβ€˜π‘) ∈ suc 𝐴))
339, 32biimtrid 241 . . . 4 (( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) β†’ (𝑐 ∈ {𝑏 ∈ 𝑋 ∣ 𝑏 β‰Ό 𝐴} β†’ (cardβ€˜π‘) ∈ suc 𝐴))
34 elrabi 3678 . . . . 5 (𝑐 ∈ {𝑏 ∈ 𝑋 ∣ 𝑏 β‰Ό 𝐴} β†’ 𝑐 ∈ 𝑋)
35 elrabi 3678 . . . . 5 (𝑑 ∈ {𝑏 ∈ 𝑋 ∣ 𝑏 β‰Ό 𝐴} β†’ 𝑑 ∈ 𝑋)
36 ssel 3976 . . . . . . . . . . 11 (𝑋 βŠ† Fin β†’ (𝑐 ∈ 𝑋 β†’ 𝑐 ∈ Fin))
37 ssel 3976 . . . . . . . . . . 11 (𝑋 βŠ† Fin β†’ (𝑑 ∈ 𝑋 β†’ 𝑑 ∈ Fin))
3836, 37anim12d 610 . . . . . . . . . 10 (𝑋 βŠ† Fin β†’ ((𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋) β†’ (𝑐 ∈ Fin ∧ 𝑑 ∈ Fin)))
3938imp 408 . . . . . . . . 9 ((𝑋 βŠ† Fin ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) β†’ (𝑐 ∈ Fin ∧ 𝑑 ∈ Fin))
40393ad2antl2 1187 . . . . . . . 8 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) β†’ (𝑐 ∈ Fin ∧ 𝑑 ∈ Fin))
41 sorpssi 7719 . . . . . . . . 9 (( [⊊] Or 𝑋 ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) β†’ (𝑐 βŠ† 𝑑 ∨ 𝑑 βŠ† 𝑐))
42413ad2antl1 1186 . . . . . . . 8 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) β†’ (𝑐 βŠ† 𝑑 ∨ 𝑑 βŠ† 𝑐))
43 finnum 9943 . . . . . . . . . . 11 (𝑑 ∈ Fin β†’ 𝑑 ∈ dom card)
44 carden2 9982 . . . . . . . . . . 11 ((𝑐 ∈ dom card ∧ 𝑑 ∈ dom card) β†’ ((cardβ€˜π‘) = (cardβ€˜π‘‘) ↔ 𝑐 β‰ˆ 𝑑))
4514, 43, 44syl2an 597 . . . . . . . . . 10 ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) β†’ ((cardβ€˜π‘) = (cardβ€˜π‘‘) ↔ 𝑐 β‰ˆ 𝑑))
4645adantr 482 . . . . . . . . 9 (((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (𝑐 βŠ† 𝑑 ∨ 𝑑 βŠ† 𝑐)) β†’ ((cardβ€˜π‘) = (cardβ€˜π‘‘) ↔ 𝑐 β‰ˆ 𝑑))
47 fin23lem25 10319 . . . . . . . . . . 11 ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin ∧ (𝑐 βŠ† 𝑑 ∨ 𝑑 βŠ† 𝑐)) β†’ (𝑐 β‰ˆ 𝑑 ↔ 𝑐 = 𝑑))
48473expa 1119 . . . . . . . . . 10 (((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (𝑐 βŠ† 𝑑 ∨ 𝑑 βŠ† 𝑐)) β†’ (𝑐 β‰ˆ 𝑑 ↔ 𝑐 = 𝑑))
4948biimpd 228 . . . . . . . . 9 (((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (𝑐 βŠ† 𝑑 ∨ 𝑑 βŠ† 𝑐)) β†’ (𝑐 β‰ˆ 𝑑 β†’ 𝑐 = 𝑑))
5046, 49sylbid 239 . . . . . . . 8 (((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (𝑐 βŠ† 𝑑 ∨ 𝑑 βŠ† 𝑐)) β†’ ((cardβ€˜π‘) = (cardβ€˜π‘‘) β†’ 𝑐 = 𝑑))
5140, 42, 50syl2anc 585 . . . . . . 7 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) β†’ ((cardβ€˜π‘) = (cardβ€˜π‘‘) β†’ 𝑐 = 𝑑))
52 fveq2 6892 . . . . . . 7 (𝑐 = 𝑑 β†’ (cardβ€˜π‘) = (cardβ€˜π‘‘))
5351, 52impbid1 224 . . . . . 6 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) β†’ ((cardβ€˜π‘) = (cardβ€˜π‘‘) ↔ 𝑐 = 𝑑))
5453ex 414 . . . . 5 (( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) β†’ ((𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋) β†’ ((cardβ€˜π‘) = (cardβ€˜π‘‘) ↔ 𝑐 = 𝑑)))
5534, 35, 54syl2ani 608 . . . 4 (( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) β†’ ((𝑐 ∈ {𝑏 ∈ 𝑋 ∣ 𝑏 β‰Ό 𝐴} ∧ 𝑑 ∈ {𝑏 ∈ 𝑋 ∣ 𝑏 β‰Ό 𝐴}) β†’ ((cardβ€˜π‘) = (cardβ€˜π‘‘) ↔ 𝑐 = 𝑑)))
5633, 55dom2d 8989 . . 3 (( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) β†’ (suc 𝐴 ∈ Ο‰ β†’ {𝑏 ∈ 𝑋 ∣ 𝑏 β‰Ό 𝐴} β‰Ό suc 𝐴))
577, 56mpd 15 . 2 (( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) β†’ {𝑏 ∈ 𝑋 ∣ 𝑏 β‰Ό 𝐴} β‰Ό suc 𝐴)
58 domfi 9192 . 2 ((suc 𝐴 ∈ Fin ∧ {𝑏 ∈ 𝑋 ∣ 𝑏 β‰Ό 𝐴} β‰Ό suc 𝐴) β†’ {𝑏 ∈ 𝑋 ∣ 𝑏 β‰Ό 𝐴} ∈ Fin)
596, 57, 58syl2anc 585 1 (( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) β†’ {𝑏 ∈ 𝑋 ∣ 𝑏 β‰Ό 𝐴} ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {crab 3433   ∩ cin 3948   βŠ† wss 3949   class class class wbr 5149   Or wor 5588  dom cdm 5677  Oncon0 6365  suc csuc 6367  β€˜cfv 6544   [⊊] crpss 7712  Ο‰com 7855   β‰ˆ cen 8936   β‰Ό cdom 8937  Fincfn 8939  cardccrd 9930
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  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-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  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-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-rpss 7713  df-om 7856  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934
This theorem is referenced by:  fin1a2lem11  10405
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