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Theorem alephinit 10094
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 10091 . . . . 5 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) ↔ 𝐴 ∈ ran β„΅)
21bicomi 223 . . . 4 (𝐴 ∈ ran β„΅ ↔ (Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴))
32baib 535 . . 3 (Ο‰ βŠ† 𝐴 β†’ (𝐴 ∈ ran β„΅ ↔ (cardβ€˜π΄) = 𝐴))
43adantl 481 . 2 ((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) β†’ (𝐴 ∈ ran β„΅ ↔ (cardβ€˜π΄) = 𝐴))
5 onenon 9948 . . . . . . . 8 (𝐴 ∈ On β†’ 𝐴 ∈ dom card)
65adantr 480 . . . . . . 7 ((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) β†’ 𝐴 ∈ dom card)
7 onenon 9948 . . . . . . 7 (π‘₯ ∈ On β†’ π‘₯ ∈ dom card)
8 carddom2 9976 . . . . . . 7 ((𝐴 ∈ dom card ∧ π‘₯ ∈ dom card) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π‘₯) ↔ 𝐴 β‰Ό π‘₯))
96, 7, 8syl2an 595 . . . . . 6 (((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) ∧ π‘₯ ∈ On) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π‘₯) ↔ 𝐴 β‰Ό π‘₯))
10 cardonle 9956 . . . . . . . 8 (π‘₯ ∈ On β†’ (cardβ€˜π‘₯) βŠ† π‘₯)
1110adantl 481 . . . . . . 7 (((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) ∧ π‘₯ ∈ On) β†’ (cardβ€˜π‘₯) βŠ† π‘₯)
12 sstr 3990 . . . . . . . 8 (((cardβ€˜π΄) βŠ† (cardβ€˜π‘₯) ∧ (cardβ€˜π‘₯) βŠ† π‘₯) β†’ (cardβ€˜π΄) βŠ† π‘₯)
1312expcom 413 . . . . . . 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 260 . . . . 5 (((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) ∧ π‘₯ ∈ On) β†’ (𝐴 β‰Ό π‘₯ β†’ (cardβ€˜π΄) βŠ† π‘₯))
16 sseq1 4007 . . . . . 6 ((cardβ€˜π΄) = 𝐴 β†’ ((cardβ€˜π΄) βŠ† π‘₯ ↔ 𝐴 βŠ† π‘₯))
1716imbi2d 340 . . . . 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 3153 . . 3 ((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) β†’ ((cardβ€˜π΄) = 𝐴 β†’ βˆ€π‘₯ ∈ On (𝐴 β‰Ό π‘₯ β†’ 𝐴 βŠ† π‘₯)))
20 oncardid 9955 . . . . . . 7 (𝐴 ∈ On β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
21 ensym 9003 . . . . . . 7 ((cardβ€˜π΄) β‰ˆ 𝐴 β†’ 𝐴 β‰ˆ (cardβ€˜π΄))
22 endom 8979 . . . . . . 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 480 . . . . 5 ((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) β†’ 𝐴 β‰Ό (cardβ€˜π΄))
25 cardon 9943 . . . . . 6 (cardβ€˜π΄) ∈ On
26 breq2 5152 . . . . . . . 8 (π‘₯ = (cardβ€˜π΄) β†’ (𝐴 β‰Ό π‘₯ ↔ 𝐴 β‰Ό (cardβ€˜π΄)))
27 sseq2 4008 . . . . . . . 8 (π‘₯ = (cardβ€˜π΄) β†’ (𝐴 βŠ† π‘₯ ↔ 𝐴 βŠ† (cardβ€˜π΄)))
2826, 27imbi12d 344 . . . . . . 7 (π‘₯ = (cardβ€˜π΄) β†’ ((𝐴 β‰Ό π‘₯ β†’ 𝐴 βŠ† π‘₯) ↔ (𝐴 β‰Ό (cardβ€˜π΄) β†’ 𝐴 βŠ† (cardβ€˜π΄))))
2928rspcv 3608 . . . . . 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 9956 . . . . . . 7 (𝐴 ∈ On β†’ (cardβ€˜π΄) βŠ† 𝐴)
3332adantr 480 . . . . . 6 ((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) β†’ (cardβ€˜π΄) βŠ† 𝐴)
3433biantrurd 532 . . . . 5 ((𝐴 ∈ On ∧ Ο‰ βŠ† 𝐴) β†’ (𝐴 βŠ† (cardβ€˜π΄) ↔ ((cardβ€˜π΄) βŠ† 𝐴 ∧ 𝐴 βŠ† (cardβ€˜π΄))))
35 eqss 3997 . . . . 5 ((cardβ€˜π΄) = 𝐴 ↔ ((cardβ€˜π΄) βŠ† 𝐴 ∧ 𝐴 βŠ† (cardβ€˜π΄)))
3634, 35bitr4di 289 . . . 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 279 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 395   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060   βŠ† wss 3948   class class class wbr 5148  dom cdm 5676  ran crn 5677  Oncon0 6364  β€˜cfv 6543  Ο‰com 7859   β‰ˆ cen 8940   β‰Ό cdom 8941  cardccrd 9934  β„΅cale 9935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9640
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-oi 9509  df-har 9556  df-card 9938  df-aleph 9939
This theorem is referenced by: (None)
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