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Theorem fin1a2lem9 9174
Description: Lemma for fin1a2 9181. In a chain of finite sets, initial segments are finite. (Contributed by Stefan O'Rear, 8-Nov-2014.)
Assertion
Ref Expression
fin1a2lem9 (( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) → {𝑏𝑋𝑏𝐴} ∈ 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 8096 . . . . 5 ω = (On ∩ Fin)
2 inss2 3812 . . . . 5 (On ∩ Fin) ⊆ Fin
31, 2eqsstri 3614 . . . 4 ω ⊆ Fin
4 peano2 7033 . . . 4 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
53, 4sseldi 3581 . . 3 (𝐴 ∈ ω → suc 𝐴 ∈ Fin)
653ad2ant3 1082 . 2 (( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) → suc 𝐴 ∈ Fin)
743ad2ant3 1082 . . 3 (( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) → suc 𝐴 ∈ ω)
8 breq1 4616 . . . . . 6 (𝑏 = 𝑐 → (𝑏𝐴𝑐𝐴))
98elrab 3346 . . . . 5 (𝑐 ∈ {𝑏𝑋𝑏𝐴} ↔ (𝑐𝑋𝑐𝐴))
10 simprr 795 . . . . . . . 8 ((( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) ∧ (𝑐𝑋𝑐𝐴)) → 𝑐𝐴)
11 simpl2 1063 . . . . . . . . . . 11 ((( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) ∧ (𝑐𝑋𝑐𝐴)) → 𝑋 ⊆ Fin)
12 simprl 793 . . . . . . . . . . 11 ((( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) ∧ (𝑐𝑋𝑐𝐴)) → 𝑐𝑋)
1311, 12sseldd 3584 . . . . . . . . . 10 ((( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) ∧ (𝑐𝑋𝑐𝐴)) → 𝑐 ∈ Fin)
14 finnum 8718 . . . . . . . . . 10 (𝑐 ∈ Fin → 𝑐 ∈ dom card)
1513, 14syl 17 . . . . . . . . 9 ((( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) ∧ (𝑐𝑋𝑐𝐴)) → 𝑐 ∈ dom card)
16 simpl3 1064 . . . . . . . . . . 11 ((( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) ∧ (𝑐𝑋𝑐𝐴)) → 𝐴 ∈ ω)
173, 16sseldi 3581 . . . . . . . . . 10 ((( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) ∧ (𝑐𝑋𝑐𝐴)) → 𝐴 ∈ Fin)
18 finnum 8718 . . . . . . . . . 10 (𝐴 ∈ Fin → 𝐴 ∈ dom card)
1917, 18syl 17 . . . . . . . . 9 ((( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) ∧ (𝑐𝑋𝑐𝐴)) → 𝐴 ∈ dom card)
20 carddom2 8747 . . . . . . . . 9 ((𝑐 ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘𝑐) ⊆ (card‘𝐴) ↔ 𝑐𝐴))
2115, 19, 20syl2anc 692 . . . . . . . 8 ((( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) ∧ (𝑐𝑋𝑐𝐴)) → ((card‘𝑐) ⊆ (card‘𝐴) ↔ 𝑐𝐴))
2210, 21mpbird 247 . . . . . . 7 ((( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) ∧ (𝑐𝑋𝑐𝐴)) → (card‘𝑐) ⊆ (card‘𝐴))
2322ex 450 . . . . . 6 (( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) → ((𝑐𝑋𝑐𝐴) → (card‘𝑐) ⊆ (card‘𝐴)))
24 cardnn 8733 . . . . . . . . 9 (𝐴 ∈ ω → (card‘𝐴) = 𝐴)
2524sseq2d 3612 . . . . . . . 8 (𝐴 ∈ ω → ((card‘𝑐) ⊆ (card‘𝐴) ↔ (card‘𝑐) ⊆ 𝐴))
26 cardon 8714 . . . . . . . . 9 (card‘𝑐) ∈ On
27 nnon 7018 . . . . . . . . 9 (𝐴 ∈ ω → 𝐴 ∈ On)
28 onsssuc 5772 . . . . . . . . 9 (((card‘𝑐) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝑐) ⊆ 𝐴 ↔ (card‘𝑐) ∈ suc 𝐴))
2926, 27, 28sylancr 694 . . . . . . . 8 (𝐴 ∈ ω → ((card‘𝑐) ⊆ 𝐴 ↔ (card‘𝑐) ∈ suc 𝐴))
3025, 29bitrd 268 . . . . . . 7 (𝐴 ∈ ω → ((card‘𝑐) ⊆ (card‘𝐴) ↔ (card‘𝑐) ∈ suc 𝐴))
31303ad2ant3 1082 . . . . . 6 (( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) → ((card‘𝑐) ⊆ (card‘𝐴) ↔ (card‘𝑐) ∈ suc 𝐴))
3223, 31sylibd 229 . . . . 5 (( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) → ((𝑐𝑋𝑐𝐴) → (card‘𝑐) ∈ suc 𝐴))
339, 32syl5bi 232 . . . 4 (( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) → (𝑐 ∈ {𝑏𝑋𝑏𝐴} → (card‘𝑐) ∈ suc 𝐴))
34 elrabi 3342 . . . . 5 (𝑐 ∈ {𝑏𝑋𝑏𝐴} → 𝑐𝑋)
35 elrabi 3342 . . . . 5 (𝑑 ∈ {𝑏𝑋𝑏𝐴} → 𝑑𝑋)
36 ssel 3577 . . . . . . . . . . 11 (𝑋 ⊆ Fin → (𝑐𝑋𝑐 ∈ Fin))
37 ssel 3577 . . . . . . . . . . 11 (𝑋 ⊆ Fin → (𝑑𝑋𝑑 ∈ Fin))
3836, 37anim12d 585 . . . . . . . . . 10 (𝑋 ⊆ Fin → ((𝑐𝑋𝑑𝑋) → (𝑐 ∈ Fin ∧ 𝑑 ∈ Fin)))
3938imp 445 . . . . . . . . 9 ((𝑋 ⊆ Fin ∧ (𝑐𝑋𝑑𝑋)) → (𝑐 ∈ Fin ∧ 𝑑 ∈ Fin))
40393ad2antl2 1222 . . . . . . . 8 ((( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) ∧ (𝑐𝑋𝑑𝑋)) → (𝑐 ∈ Fin ∧ 𝑑 ∈ Fin))
41 sorpssi 6896 . . . . . . . . 9 (( [] Or 𝑋 ∧ (𝑐𝑋𝑑𝑋)) → (𝑐𝑑𝑑𝑐))
42413ad2antl1 1221 . . . . . . . 8 ((( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) ∧ (𝑐𝑋𝑑𝑋)) → (𝑐𝑑𝑑𝑐))
43 finnum 8718 . . . . . . . . . . 11 (𝑑 ∈ Fin → 𝑑 ∈ dom card)
44 carden2 8757 . . . . . . . . . . 11 ((𝑐 ∈ dom card ∧ 𝑑 ∈ dom card) → ((card‘𝑐) = (card‘𝑑) ↔ 𝑐𝑑))
4514, 43, 44syl2an 494 . . . . . . . . . 10 ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) → ((card‘𝑐) = (card‘𝑑) ↔ 𝑐𝑑))
4645adantr 481 . . . . . . . . 9 (((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (𝑐𝑑𝑑𝑐)) → ((card‘𝑐) = (card‘𝑑) ↔ 𝑐𝑑))
47 fin23lem25 9090 . . . . . . . . . . 11 ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin ∧ (𝑐𝑑𝑑𝑐)) → (𝑐𝑑𝑐 = 𝑑))
48473expa 1262 . . . . . . . . . 10 (((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (𝑐𝑑𝑑𝑐)) → (𝑐𝑑𝑐 = 𝑑))
4948biimpd 219 . . . . . . . . 9 (((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (𝑐𝑑𝑑𝑐)) → (𝑐𝑑𝑐 = 𝑑))
5046, 49sylbid 230 . . . . . . . 8 (((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (𝑐𝑑𝑑𝑐)) → ((card‘𝑐) = (card‘𝑑) → 𝑐 = 𝑑))
5140, 42, 50syl2anc 692 . . . . . . 7 ((( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) ∧ (𝑐𝑋𝑑𝑋)) → ((card‘𝑐) = (card‘𝑑) → 𝑐 = 𝑑))
52 fveq2 6148 . . . . . . 7 (𝑐 = 𝑑 → (card‘𝑐) = (card‘𝑑))
5351, 52impbid1 215 . . . . . 6 ((( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) ∧ (𝑐𝑋𝑑𝑋)) → ((card‘𝑐) = (card‘𝑑) ↔ 𝑐 = 𝑑))
5453ex 450 . . . . 5 (( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) → ((𝑐𝑋𝑑𝑋) → ((card‘𝑐) = (card‘𝑑) ↔ 𝑐 = 𝑑)))
5534, 35, 54syl2ani 687 . . . 4 (( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) → ((𝑐 ∈ {𝑏𝑋𝑏𝐴} ∧ 𝑑 ∈ {𝑏𝑋𝑏𝐴}) → ((card‘𝑐) = (card‘𝑑) ↔ 𝑐 = 𝑑)))
5633, 55dom2d 7940 . . 3 (( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) → (suc 𝐴 ∈ ω → {𝑏𝑋𝑏𝐴} ≼ suc 𝐴))
577, 56mpd 15 . 2 (( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) → {𝑏𝑋𝑏𝐴} ≼ suc 𝐴)
58 domfi 8125 . 2 ((suc 𝐴 ∈ Fin ∧ {𝑏𝑋𝑏𝐴} ≼ suc 𝐴) → {𝑏𝑋𝑏𝐴} ∈ Fin)
596, 57, 58syl2anc 692 1 (( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) → {𝑏𝑋𝑏𝐴} ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  {crab 2911  cin 3554  wss 3555   class class class wbr 4613   Or wor 4994  dom cdm 5074  Oncon0 5682  suc csuc 5684  cfv 5847   [] crpss 6889  ωcom 7012  cen 7896  cdom 7897  Fincfn 7899  cardccrd 8705
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-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-rpss 6890  df-om 7013  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709
This theorem is referenced by:  fin1a2lem11  9176
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