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Theorem carduni 8662
Description: The union of a set of cardinals is a cardinal. Theorem 18.14 of [Monk1] p. 133. (Contributed by Mario Carneiro, 20-Jan-2013.)
Assertion
Ref Expression
carduni (𝐴𝑉 → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → (card‘ 𝐴) = 𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem carduni
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6083 . . . . . . . . . 10 (𝑥 = 𝑦 → (card‘𝑥) = (card‘𝑦))
2 id 22 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
31, 2eqeq12d 2619 . . . . . . . . 9 (𝑥 = 𝑦 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑦) = 𝑦))
43rspcv 3272 . . . . . . . 8 (𝑦𝐴 → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → (card‘𝑦) = 𝑦))
5 cardon 8625 . . . . . . . . 9 (card‘𝑦) ∈ On
6 eleq1 2670 . . . . . . . . 9 ((card‘𝑦) = 𝑦 → ((card‘𝑦) ∈ On ↔ 𝑦 ∈ On))
75, 6mpbii 221 . . . . . . . 8 ((card‘𝑦) = 𝑦𝑦 ∈ On)
84, 7syl6com 36 . . . . . . 7 (∀𝑥𝐴 (card‘𝑥) = 𝑥 → (𝑦𝐴𝑦 ∈ On))
98ssrdv 3568 . . . . . 6 (∀𝑥𝐴 (card‘𝑥) = 𝑥𝐴 ⊆ On)
10 ssonuni 6850 . . . . . 6 (𝐴𝑉 → (𝐴 ⊆ On → 𝐴 ∈ On))
119, 10syl5 33 . . . . 5 (𝐴𝑉 → (∀𝑥𝐴 (card‘𝑥) = 𝑥 𝐴 ∈ On))
1211imp 443 . . . 4 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → 𝐴 ∈ On)
13 cardonle 8638 . . . 4 ( 𝐴 ∈ On → (card‘ 𝐴) ⊆ 𝐴)
1412, 13syl 17 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → (card‘ 𝐴) ⊆ 𝐴)
15 cardon 8625 . . . . 5 (card‘ 𝐴) ∈ On
1615onirri 5732 . . . 4 ¬ (card‘ 𝐴) ∈ (card‘ 𝐴)
17 eluni 4364 . . . . . . . 8 ((card‘ 𝐴) ∈ 𝐴 ↔ ∃𝑦((card‘ 𝐴) ∈ 𝑦𝑦𝐴))
18 elssuni 4392 . . . . . . . . . . . . . . . . . 18 (𝑦𝐴𝑦 𝐴)
19 ssdomg 7859 . . . . . . . . . . . . . . . . . . 19 ( 𝐴 ∈ On → (𝑦 𝐴𝑦 𝐴))
2019adantl 480 . . . . . . . . . . . . . . . . . 18 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → (𝑦 𝐴𝑦 𝐴))
2118, 20syl5 33 . . . . . . . . . . . . . . . . 17 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → (𝑦𝐴𝑦 𝐴))
22 id 22 . . . . . . . . . . . . . . . . . . 19 ((card‘𝑦) = 𝑦 → (card‘𝑦) = 𝑦)
23 onenon 8630 . . . . . . . . . . . . . . . . . . . 20 ((card‘𝑦) ∈ On → (card‘𝑦) ∈ dom card)
245, 23ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (card‘𝑦) ∈ dom card
2522, 24syl6eqelr 2691 . . . . . . . . . . . . . . . . . 18 ((card‘𝑦) = 𝑦𝑦 ∈ dom card)
26 onenon 8630 . . . . . . . . . . . . . . . . . 18 ( 𝐴 ∈ On → 𝐴 ∈ dom card)
27 carddom2 8658 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘𝑦) ⊆ (card‘ 𝐴) ↔ 𝑦 𝐴))
2825, 26, 27syl2an 492 . . . . . . . . . . . . . . . . 17 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → ((card‘𝑦) ⊆ (card‘ 𝐴) ↔ 𝑦 𝐴))
2921, 28sylibrd 247 . . . . . . . . . . . . . . . 16 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → (𝑦𝐴 → (card‘𝑦) ⊆ (card‘ 𝐴)))
30 sseq1 3583 . . . . . . . . . . . . . . . . 17 ((card‘𝑦) = 𝑦 → ((card‘𝑦) ⊆ (card‘ 𝐴) ↔ 𝑦 ⊆ (card‘ 𝐴)))
3130adantr 479 . . . . . . . . . . . . . . . 16 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → ((card‘𝑦) ⊆ (card‘ 𝐴) ↔ 𝑦 ⊆ (card‘ 𝐴)))
3229, 31sylibd 227 . . . . . . . . . . . . . . 15 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → (𝑦𝐴𝑦 ⊆ (card‘ 𝐴)))
33 ssel 3556 . . . . . . . . . . . . . . 15 (𝑦 ⊆ (card‘ 𝐴) → ((card‘ 𝐴) ∈ 𝑦 → (card‘ 𝐴) ∈ (card‘ 𝐴)))
3432, 33syl6 34 . . . . . . . . . . . . . 14 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → (𝑦𝐴 → ((card‘ 𝐴) ∈ 𝑦 → (card‘ 𝐴) ∈ (card‘ 𝐴))))
3534ex 448 . . . . . . . . . . . . 13 ((card‘𝑦) = 𝑦 → ( 𝐴 ∈ On → (𝑦𝐴 → ((card‘ 𝐴) ∈ 𝑦 → (card‘ 𝐴) ∈ (card‘ 𝐴)))))
3635com3r 84 . . . . . . . . . . . 12 (𝑦𝐴 → ((card‘𝑦) = 𝑦 → ( 𝐴 ∈ On → ((card‘ 𝐴) ∈ 𝑦 → (card‘ 𝐴) ∈ (card‘ 𝐴)))))
374, 36syld 45 . . . . . . . . . . 11 (𝑦𝐴 → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → ( 𝐴 ∈ On → ((card‘ 𝐴) ∈ 𝑦 → (card‘ 𝐴) ∈ (card‘ 𝐴)))))
3837com4r 91 . . . . . . . . . 10 ((card‘ 𝐴) ∈ 𝑦 → (𝑦𝐴 → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → ( 𝐴 ∈ On → (card‘ 𝐴) ∈ (card‘ 𝐴)))))
3938imp 443 . . . . . . . . 9 (((card‘ 𝐴) ∈ 𝑦𝑦𝐴) → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → ( 𝐴 ∈ On → (card‘ 𝐴) ∈ (card‘ 𝐴))))
4039exlimiv 1843 . . . . . . . 8 (∃𝑦((card‘ 𝐴) ∈ 𝑦𝑦𝐴) → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → ( 𝐴 ∈ On → (card‘ 𝐴) ∈ (card‘ 𝐴))))
4117, 40sylbi 205 . . . . . . 7 ((card‘ 𝐴) ∈ 𝐴 → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → ( 𝐴 ∈ On → (card‘ 𝐴) ∈ (card‘ 𝐴))))
4241com13 85 . . . . . 6 ( 𝐴 ∈ On → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → ((card‘ 𝐴) ∈ 𝐴 → (card‘ 𝐴) ∈ (card‘ 𝐴))))
4342imp 443 . . . . 5 (( 𝐴 ∈ On ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → ((card‘ 𝐴) ∈ 𝐴 → (card‘ 𝐴) ∈ (card‘ 𝐴)))
4412, 43sylancom 697 . . . 4 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → ((card‘ 𝐴) ∈ 𝐴 → (card‘ 𝐴) ∈ (card‘ 𝐴)))
4516, 44mtoi 188 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → ¬ (card‘ 𝐴) ∈ 𝐴)
4615onordi 5730 . . . 4 Ord (card‘ 𝐴)
47 eloni 5631 . . . . 5 ( 𝐴 ∈ On → Ord 𝐴)
4812, 47syl 17 . . . 4 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → Ord 𝐴)
49 ordtri4 5659 . . . 4 ((Ord (card‘ 𝐴) ∧ Ord 𝐴) → ((card‘ 𝐴) = 𝐴 ↔ ((card‘ 𝐴) ⊆ 𝐴 ∧ ¬ (card‘ 𝐴) ∈ 𝐴)))
5046, 48, 49sylancr 693 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → ((card‘ 𝐴) = 𝐴 ↔ ((card‘ 𝐴) ⊆ 𝐴 ∧ ¬ (card‘ 𝐴) ∈ 𝐴)))
5114, 45, 50mpbir2and 958 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → (card‘ 𝐴) = 𝐴)
5251ex 448 1 (𝐴𝑉 → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → (card‘ 𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1975  wral 2890  wss 3534   cuni 4361   class class class wbr 4572  dom cdm 5023  Ord word 5620  Oncon0 5621  cfv 5785  cdom 7811  cardccrd 8616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-ord 5624  df-on 5625  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-er 7601  df-en 7814  df-dom 7815  df-sdom 7816  df-card 8620
This theorem is referenced by:  cardiun  8663  carduniima  8774
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