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Theorem alephinit 10092
Description: An infinite initial ordinal is characterized by the property of being initial - that is, it is a subset of any dominating ordinal. (Contributed by Jeff Hankins, 29-Oct-2009.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
alephinit ((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) β†’ (𝐴 ∈ ran β„΅ ↔ βˆ€π‘₯ ∈ On (𝐴 β‰Ό π‘₯ β†’ 𝐴 βŠ† π‘₯)))
Distinct variable group:   π‘₯,𝐴

Proof of Theorem alephinit
StepHypRef Expression
1 isinfcard 10089 . . . . 5 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) ↔ 𝐴 ∈ ran β„΅)
21bicomi 223 . . . 4 (𝐴 ∈ ran β„΅ ↔ (Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴))
32baib 534 . . 3 (Ο‰ βŠ† 𝐴 β†’ (𝐴 ∈ ran β„΅ ↔ (cardβ€˜π΄) = 𝐴))
43adantl 480 . 2 ((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) β†’ (𝐴 ∈ ran β„΅ ↔ (cardβ€˜π΄) = 𝐴))
5 onenon 9946 . . . . . . . 8 (𝐴 ∈ On β†’ 𝐴 ∈ dom card)
65adantr 479 . . . . . . 7 ((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) β†’ 𝐴 ∈ dom card)
7 onenon 9946 . . . . . . 7 (π‘₯ ∈ On β†’ π‘₯ ∈ dom card)
8 carddom2 9974 . . . . . . 7 ((𝐴 ∈ dom card ∧ π‘₯ ∈ dom card) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π‘₯) ↔ 𝐴 β‰Ό π‘₯))
96, 7, 8syl2an 594 . . . . . 6 (((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) ∧ π‘₯ ∈ On) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π‘₯) ↔ 𝐴 β‰Ό π‘₯))
10 cardonle 9954 . . . . . . . 8 (π‘₯ ∈ On β†’ (cardβ€˜π‘₯) βŠ† π‘₯)
1110adantl 480 . . . . . . 7 (((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) ∧ π‘₯ ∈ On) β†’ (cardβ€˜π‘₯) βŠ† π‘₯)
12 sstr 3989 . . . . . . . 8 (((cardβ€˜π΄) βŠ† (cardβ€˜π‘₯) ∧ (cardβ€˜π‘₯) βŠ† π‘₯) β†’ (cardβ€˜π΄) βŠ† π‘₯)
1312expcom 412 . . . . . . 7 ((cardβ€˜π‘₯) βŠ† π‘₯ β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π‘₯) β†’ (cardβ€˜π΄) βŠ† π‘₯))
1411, 13syl 17 . . . . . 6 (((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) ∧ π‘₯ ∈ On) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π‘₯) β†’ (cardβ€˜π΄) βŠ† π‘₯))
159, 14sylbird 259 . . . . 5 (((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) ∧ π‘₯ ∈ On) β†’ (𝐴 β‰Ό π‘₯ β†’ (cardβ€˜π΄) βŠ† π‘₯))
16 sseq1 4006 . . . . . 6 ((cardβ€˜π΄) = 𝐴 β†’ ((cardβ€˜π΄) βŠ† π‘₯ ↔ 𝐴 βŠ† π‘₯))
1716imbi2d 339 . . . . 5 ((cardβ€˜π΄) = 𝐴 β†’ ((𝐴 β‰Ό π‘₯ β†’ (cardβ€˜π΄) βŠ† π‘₯) ↔ (𝐴 β‰Ό π‘₯ β†’ 𝐴 βŠ† π‘₯)))
1815, 17syl5ibcom 244 . . . 4 (((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) ∧ π‘₯ ∈ On) β†’ ((cardβ€˜π΄) = 𝐴 β†’ (𝐴 β‰Ό π‘₯ β†’ 𝐴 βŠ† π‘₯)))
1918ralrimdva 3152 . . 3 ((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) β†’ ((cardβ€˜π΄) = 𝐴 β†’ βˆ€π‘₯ ∈ On (𝐴 β‰Ό π‘₯ β†’ 𝐴 βŠ† π‘₯)))
20 oncardid 9953 . . . . . . 7 (𝐴 ∈ On β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
21 ensym 9001 . . . . . . 7 ((cardβ€˜π΄) β‰ˆ 𝐴 β†’ 𝐴 β‰ˆ (cardβ€˜π΄))
22 endom 8977 . . . . . . 7 (𝐴 β‰ˆ (cardβ€˜π΄) β†’ 𝐴 β‰Ό (cardβ€˜π΄))
2320, 21, 223syl 18 . . . . . 6 (𝐴 ∈ On β†’ 𝐴 β‰Ό (cardβ€˜π΄))
2423adantr 479 . . . . 5 ((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) β†’ 𝐴 β‰Ό (cardβ€˜π΄))
25 cardon 9941 . . . . . 6 (cardβ€˜π΄) ∈ On
26 breq2 5151 . . . . . . . 8 (π‘₯ = (cardβ€˜π΄) β†’ (𝐴 β‰Ό π‘₯ ↔ 𝐴 β‰Ό (cardβ€˜π΄)))
27 sseq2 4007 . . . . . . . 8 (π‘₯ = (cardβ€˜π΄) β†’ (𝐴 βŠ† π‘₯ ↔ 𝐴 βŠ† (cardβ€˜π΄)))
2826, 27imbi12d 343 . . . . . . 7 (π‘₯ = (cardβ€˜π΄) β†’ ((𝐴 β‰Ό π‘₯ β†’ 𝐴 βŠ† π‘₯) ↔ (𝐴 β‰Ό (cardβ€˜π΄) β†’ 𝐴 βŠ† (cardβ€˜π΄))))
2928rspcv 3607 . . . . . 6 ((cardβ€˜π΄) ∈ On β†’ (βˆ€π‘₯ ∈ On (𝐴 β‰Ό π‘₯ β†’ 𝐴 βŠ† π‘₯) β†’ (𝐴 β‰Ό (cardβ€˜π΄) β†’ 𝐴 βŠ† (cardβ€˜π΄))))
3025, 29ax-mp 5 . . . . 5 (βˆ€π‘₯ ∈ On (𝐴 β‰Ό π‘₯ β†’ 𝐴 βŠ† π‘₯) β†’ (𝐴 β‰Ό (cardβ€˜π΄) β†’ 𝐴 βŠ† (cardβ€˜π΄)))
3124, 30syl5com 31 . . . 4 ((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) β†’ (βˆ€π‘₯ ∈ On (𝐴 β‰Ό π‘₯ β†’ 𝐴 βŠ† π‘₯) β†’ 𝐴 βŠ† (cardβ€˜π΄)))
32 cardonle 9954 . . . . . . 7 (𝐴 ∈ On β†’ (cardβ€˜π΄) βŠ† 𝐴)
3332adantr 479 . . . . . 6 ((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) β†’ (cardβ€˜π΄) βŠ† 𝐴)
3433biantrurd 531 . . . . 5 ((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) β†’ (𝐴 βŠ† (cardβ€˜π΄) ↔ ((cardβ€˜π΄) βŠ† 𝐴 ∧ 𝐴 βŠ† (cardβ€˜π΄))))
35 eqss 3996 . . . . 5 ((cardβ€˜π΄) = 𝐴 ↔ ((cardβ€˜π΄) βŠ† 𝐴 ∧ 𝐴 βŠ† (cardβ€˜π΄)))
3634, 35bitr4di 288 . . . 4 ((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) β†’ (𝐴 βŠ† (cardβ€˜π΄) ↔ (cardβ€˜π΄) = 𝐴))
3731, 36sylibd 238 . . 3 ((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) β†’ (βˆ€π‘₯ ∈ On (𝐴 β‰Ό π‘₯ β†’ 𝐴 βŠ† π‘₯) β†’ (cardβ€˜π΄) = 𝐴))
3819, 37impbid 211 . 2 ((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) β†’ ((cardβ€˜π΄) = 𝐴 ↔ βˆ€π‘₯ ∈ On (𝐴 β‰Ό π‘₯ β†’ 𝐴 βŠ† π‘₯)))
394, 38bitrd 278 1 ((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) β†’ (𝐴 ∈ ran β„΅ ↔ βˆ€π‘₯ ∈ On (𝐴 β‰Ό π‘₯ β†’ 𝐴 βŠ† π‘₯)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059   βŠ† wss 3947   class class class wbr 5147  dom cdm 5675  ran crn 5676  Oncon0 6363  β€˜cfv 6542  Ο‰com 7857   β‰ˆ cen 8938   β‰Ό cdom 8939  cardccrd 9932  β„΅cale 9933
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  ax-inf2 9638
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-rmo 3374  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-se 5631  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-pred 6299  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-isom 6551  df-riota 7367  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-oi 9507  df-har 9554  df-card 9936  df-aleph 9937
This theorem is referenced by: (None)
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