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Theorem fin1a2lem9 10405
Description: Lemma for fin1a2 10412. 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 9233 . . . . 5 Ο‰ = (On ∩ Fin)
2 inss2 4228 . . . . 5 (On ∩ Fin) βŠ† Fin
31, 2eqsstri 4015 . . . 4 Ο‰ βŠ† Fin
4 peano2 7883 . . . 4 (𝐴 ∈ Ο‰ β†’ suc 𝐴 ∈ Ο‰)
53, 4sselid 3979 . . 3 (𝐴 ∈ Ο‰ β†’ suc 𝐴 ∈ Fin)
653ad2ant3 1133 . 2 (( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) β†’ suc 𝐴 ∈ Fin)
743ad2ant3 1133 . . 3 (( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) β†’ suc 𝐴 ∈ Ο‰)
8 breq1 5150 . . . . . 6 (𝑏 = 𝑐 β†’ (𝑏 β‰Ό 𝐴 ↔ 𝑐 β‰Ό 𝐴))
98elrab 3682 . . . . 5 (𝑐 ∈ {𝑏 ∈ 𝑋 ∣ 𝑏 β‰Ό 𝐴} ↔ (𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴))
10 simprr 769 . . . . . . . 8 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴)) β†’ 𝑐 β‰Ό 𝐴)
11 simpl2 1190 . . . . . . . . . . 11 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴)) β†’ 𝑋 βŠ† Fin)
12 simprl 767 . . . . . . . . . . 11 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴)) β†’ 𝑐 ∈ 𝑋)
1311, 12sseldd 3982 . . . . . . . . . 10 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴)) β†’ 𝑐 ∈ Fin)
14 finnum 9945 . . . . . . . . . 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 1191 . . . . . . . . . . 11 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴)) β†’ 𝐴 ∈ Ο‰)
173, 16sselid 3979 . . . . . . . . . 10 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴)) β†’ 𝐴 ∈ Fin)
18 finnum 9945 . . . . . . . . . 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 9974 . . . . . . . . 9 ((𝑐 ∈ dom card ∧ 𝐴 ∈ dom card) β†’ ((cardβ€˜π‘) βŠ† (cardβ€˜π΄) ↔ 𝑐 β‰Ό 𝐴))
2115, 19, 20syl2anc 582 . . . . . . . 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 256 . . . . . . 7 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴)) β†’ (cardβ€˜π‘) βŠ† (cardβ€˜π΄))
2322ex 411 . . . . . 6 (( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) β†’ ((𝑐 ∈ 𝑋 ∧ 𝑐 β‰Ό 𝐴) β†’ (cardβ€˜π‘) βŠ† (cardβ€˜π΄)))
24 cardnn 9960 . . . . . . . . 9 (𝐴 ∈ Ο‰ β†’ (cardβ€˜π΄) = 𝐴)
2524sseq2d 4013 . . . . . . . 8 (𝐴 ∈ Ο‰ β†’ ((cardβ€˜π‘) βŠ† (cardβ€˜π΄) ↔ (cardβ€˜π‘) βŠ† 𝐴))
26 cardon 9941 . . . . . . . . 9 (cardβ€˜π‘) ∈ On
27 nnon 7863 . . . . . . . . 9 (𝐴 ∈ Ο‰ β†’ 𝐴 ∈ On)
28 onsssuc 6453 . . . . . . . . 9 (((cardβ€˜π‘) ∈ On ∧ 𝐴 ∈ On) β†’ ((cardβ€˜π‘) βŠ† 𝐴 ↔ (cardβ€˜π‘) ∈ suc 𝐴))
2926, 27, 28sylancr 585 . . . . . . . 8 (𝐴 ∈ Ο‰ β†’ ((cardβ€˜π‘) βŠ† 𝐴 ↔ (cardβ€˜π‘) ∈ suc 𝐴))
3025, 29bitrd 278 . . . . . . 7 (𝐴 ∈ Ο‰ β†’ ((cardβ€˜π‘) βŠ† (cardβ€˜π΄) ↔ (cardβ€˜π‘) ∈ suc 𝐴))
31303ad2ant3 1133 . . . . . 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 3676 . . . . 5 (𝑐 ∈ {𝑏 ∈ 𝑋 ∣ 𝑏 β‰Ό 𝐴} β†’ 𝑐 ∈ 𝑋)
35 elrabi 3676 . . . . 5 (𝑑 ∈ {𝑏 ∈ 𝑋 ∣ 𝑏 β‰Ό 𝐴} β†’ 𝑑 ∈ 𝑋)
36 ssel 3974 . . . . . . . . . . 11 (𝑋 βŠ† Fin β†’ (𝑐 ∈ 𝑋 β†’ 𝑐 ∈ Fin))
37 ssel 3974 . . . . . . . . . . 11 (𝑋 βŠ† Fin β†’ (𝑑 ∈ 𝑋 β†’ 𝑑 ∈ Fin))
3836, 37anim12d 607 . . . . . . . . . 10 (𝑋 βŠ† Fin β†’ ((𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋) β†’ (𝑐 ∈ Fin ∧ 𝑑 ∈ Fin)))
3938imp 405 . . . . . . . . 9 ((𝑋 βŠ† Fin ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) β†’ (𝑐 ∈ Fin ∧ 𝑑 ∈ Fin))
40393ad2antl2 1184 . . . . . . . 8 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) β†’ (𝑐 ∈ Fin ∧ 𝑑 ∈ Fin))
41 sorpssi 7721 . . . . . . . . 9 (( [⊊] Or 𝑋 ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) β†’ (𝑐 βŠ† 𝑑 ∨ 𝑑 βŠ† 𝑐))
42413ad2antl1 1183 . . . . . . . 8 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) β†’ (𝑐 βŠ† 𝑑 ∨ 𝑑 βŠ† 𝑐))
43 finnum 9945 . . . . . . . . . . 11 (𝑑 ∈ Fin β†’ 𝑑 ∈ dom card)
44 carden2 9984 . . . . . . . . . . 11 ((𝑐 ∈ dom card ∧ 𝑑 ∈ dom card) β†’ ((cardβ€˜π‘) = (cardβ€˜π‘‘) ↔ 𝑐 β‰ˆ 𝑑))
4514, 43, 44syl2an 594 . . . . . . . . . 10 ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) β†’ ((cardβ€˜π‘) = (cardβ€˜π‘‘) ↔ 𝑐 β‰ˆ 𝑑))
4645adantr 479 . . . . . . . . 9 (((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (𝑐 βŠ† 𝑑 ∨ 𝑑 βŠ† 𝑐)) β†’ ((cardβ€˜π‘) = (cardβ€˜π‘‘) ↔ 𝑐 β‰ˆ 𝑑))
47 fin23lem25 10321 . . . . . . . . . . 11 ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin ∧ (𝑐 βŠ† 𝑑 ∨ 𝑑 βŠ† 𝑐)) β†’ (𝑐 β‰ˆ 𝑑 ↔ 𝑐 = 𝑑))
48473expa 1116 . . . . . . . . . 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 582 . . . . . . 7 ((( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) β†’ ((cardβ€˜π‘) = (cardβ€˜π‘‘) β†’ 𝑐 = 𝑑))
52 fveq2 6890 . . . . . . 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 411 . . . . 5 (( [⊊] Or 𝑋 ∧ 𝑋 βŠ† Fin ∧ 𝐴 ∈ Ο‰) β†’ ((𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋) β†’ ((cardβ€˜π‘) = (cardβ€˜π‘‘) ↔ 𝑐 = 𝑑)))
5534, 35, 54syl2ani 605 . . . 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 8991 . . 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 9194 . 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 582 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 394   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  {crab 3430   ∩ cin 3946   βŠ† wss 3947   class class class wbr 5147   Or wor 5586  dom cdm 5675  Oncon0 6363  suc csuc 6365  β€˜cfv 6542   [⊊] crpss 7714  Ο‰com 7857   β‰ˆ cen 8938   β‰Ό cdom 8939  Fincfn 8941  cardccrd 9932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-rpss 7715  df-om 7858  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936
This theorem is referenced by:  fin1a2lem11  10407
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