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Theorem carduni 9404
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 ssonuni 7494 . . . . 5 (𝐴𝑉 → (𝐴 ⊆ On → 𝐴 ∈ On))
2 fveq2 6669 . . . . . . . . 9 (𝑥 = 𝑦 → (card‘𝑥) = (card‘𝑦))
3 id 22 . . . . . . . . 9 (𝑥 = 𝑦𝑥 = 𝑦)
42, 3eqeq12d 2842 . . . . . . . 8 (𝑥 = 𝑦 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑦) = 𝑦))
54rspcv 3622 . . . . . . 7 (𝑦𝐴 → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → (card‘𝑦) = 𝑦))
6 cardon 9367 . . . . . . . 8 (card‘𝑦) ∈ On
7 eleq1 2905 . . . . . . . 8 ((card‘𝑦) = 𝑦 → ((card‘𝑦) ∈ On ↔ 𝑦 ∈ On))
86, 7mpbii 234 . . . . . . 7 ((card‘𝑦) = 𝑦𝑦 ∈ On)
95, 8syl6com 37 . . . . . 6 (∀𝑥𝐴 (card‘𝑥) = 𝑥 → (𝑦𝐴𝑦 ∈ On))
109ssrdv 3977 . . . . 5 (∀𝑥𝐴 (card‘𝑥) = 𝑥𝐴 ⊆ On)
111, 10impel 506 . . . 4 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → 𝐴 ∈ On)
12 cardonle 9380 . . . 4 ( 𝐴 ∈ On → (card‘ 𝐴) ⊆ 𝐴)
1311, 12syl 17 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → (card‘ 𝐴) ⊆ 𝐴)
14 cardon 9367 . . . . 5 (card‘ 𝐴) ∈ On
1514onirri 6296 . . . 4 ¬ (card‘ 𝐴) ∈ (card‘ 𝐴)
16 eluni 4840 . . . . . . . 8 ((card‘ 𝐴) ∈ 𝐴 ↔ ∃𝑦((card‘ 𝐴) ∈ 𝑦𝑦𝐴))
17 elssuni 4866 . . . . . . . . . . . . . . . . . 18 (𝑦𝐴𝑦 𝐴)
18 ssdomg 8549 . . . . . . . . . . . . . . . . . . 19 ( 𝐴 ∈ On → (𝑦 𝐴𝑦 𝐴))
1918adantl 482 . . . . . . . . . . . . . . . . . 18 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → (𝑦 𝐴𝑦 𝐴))
2017, 19syl5 34 . . . . . . . . . . . . . . . . 17 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → (𝑦𝐴𝑦 𝐴))
21 id 22 . . . . . . . . . . . . . . . . . . 19 ((card‘𝑦) = 𝑦 → (card‘𝑦) = 𝑦)
22 onenon 9372 . . . . . . . . . . . . . . . . . . . 20 ((card‘𝑦) ∈ On → (card‘𝑦) ∈ dom card)
236, 22ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (card‘𝑦) ∈ dom card
2421, 23syl6eqelr 2927 . . . . . . . . . . . . . . . . . 18 ((card‘𝑦) = 𝑦𝑦 ∈ dom card)
25 onenon 9372 . . . . . . . . . . . . . . . . . 18 ( 𝐴 ∈ On → 𝐴 ∈ dom card)
26 carddom2 9400 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘𝑦) ⊆ (card‘ 𝐴) ↔ 𝑦 𝐴))
2724, 25, 26syl2an 595 . . . . . . . . . . . . . . . . 17 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → ((card‘𝑦) ⊆ (card‘ 𝐴) ↔ 𝑦 𝐴))
2820, 27sylibrd 260 . . . . . . . . . . . . . . . 16 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → (𝑦𝐴 → (card‘𝑦) ⊆ (card‘ 𝐴)))
29 sseq1 3996 . . . . . . . . . . . . . . . . 17 ((card‘𝑦) = 𝑦 → ((card‘𝑦) ⊆ (card‘ 𝐴) ↔ 𝑦 ⊆ (card‘ 𝐴)))
3029adantr 481 . . . . . . . . . . . . . . . 16 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → ((card‘𝑦) ⊆ (card‘ 𝐴) ↔ 𝑦 ⊆ (card‘ 𝐴)))
3128, 30sylibd 240 . . . . . . . . . . . . . . 15 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → (𝑦𝐴𝑦 ⊆ (card‘ 𝐴)))
32 ssel 3965 . . . . . . . . . . . . . . 15 (𝑦 ⊆ (card‘ 𝐴) → ((card‘ 𝐴) ∈ 𝑦 → (card‘ 𝐴) ∈ (card‘ 𝐴)))
3331, 32syl6 35 . . . . . . . . . . . . . 14 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → (𝑦𝐴 → ((card‘ 𝐴) ∈ 𝑦 → (card‘ 𝐴) ∈ (card‘ 𝐴))))
3433ex 413 . . . . . . . . . . . . 13 ((card‘𝑦) = 𝑦 → ( 𝐴 ∈ On → (𝑦𝐴 → ((card‘ 𝐴) ∈ 𝑦 → (card‘ 𝐴) ∈ (card‘ 𝐴)))))
3534com3r 87 . . . . . . . . . . . 12 (𝑦𝐴 → ((card‘𝑦) = 𝑦 → ( 𝐴 ∈ On → ((card‘ 𝐴) ∈ 𝑦 → (card‘ 𝐴) ∈ (card‘ 𝐴)))))
365, 35syld 47 . . . . . . . . . . 11 (𝑦𝐴 → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → ( 𝐴 ∈ On → ((card‘ 𝐴) ∈ 𝑦 → (card‘ 𝐴) ∈ (card‘ 𝐴)))))
3736com4r 94 . . . . . . . . . 10 ((card‘ 𝐴) ∈ 𝑦 → (𝑦𝐴 → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → ( 𝐴 ∈ On → (card‘ 𝐴) ∈ (card‘ 𝐴)))))
3837imp 407 . . . . . . . . 9 (((card‘ 𝐴) ∈ 𝑦𝑦𝐴) → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → ( 𝐴 ∈ On → (card‘ 𝐴) ∈ (card‘ 𝐴))))
3938exlimiv 1924 . . . . . . . 8 (∃𝑦((card‘ 𝐴) ∈ 𝑦𝑦𝐴) → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → ( 𝐴 ∈ On → (card‘ 𝐴) ∈ (card‘ 𝐴))))
4016, 39sylbi 218 . . . . . . 7 ((card‘ 𝐴) ∈ 𝐴 → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → ( 𝐴 ∈ On → (card‘ 𝐴) ∈ (card‘ 𝐴))))
4140com13 88 . . . . . 6 ( 𝐴 ∈ On → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → ((card‘ 𝐴) ∈ 𝐴 → (card‘ 𝐴) ∈ (card‘ 𝐴))))
4241imp 407 . . . . 5 (( 𝐴 ∈ On ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → ((card‘ 𝐴) ∈ 𝐴 → (card‘ 𝐴) ∈ (card‘ 𝐴)))
4311, 42sylancom 588 . . . 4 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → ((card‘ 𝐴) ∈ 𝐴 → (card‘ 𝐴) ∈ (card‘ 𝐴)))
4415, 43mtoi 200 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → ¬ (card‘ 𝐴) ∈ 𝐴)
4514onordi 6294 . . . 4 Ord (card‘ 𝐴)
46 eloni 6200 . . . . 5 ( 𝐴 ∈ On → Ord 𝐴)
4711, 46syl 17 . . . 4 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → Ord 𝐴)
48 ordtri4 6227 . . . 4 ((Ord (card‘ 𝐴) ∧ Ord 𝐴) → ((card‘ 𝐴) = 𝐴 ↔ ((card‘ 𝐴) ⊆ 𝐴 ∧ ¬ (card‘ 𝐴) ∈ 𝐴)))
4945, 47, 48sylancr 587 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → ((card‘ 𝐴) = 𝐴 ↔ ((card‘ 𝐴) ⊆ 𝐴 ∧ ¬ (card‘ 𝐴) ∈ 𝐴)))
5013, 44, 49mpbir2and 709 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → (card‘ 𝐴) = 𝐴)
5150ex 413 1 (𝐴𝑉 → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → (card‘ 𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2107  wral 3143  wss 3940   cuni 4837   class class class wbr 5063  dom cdm 5554  Ord word 6189  Oncon0 6190  cfv 6354  cdom 8501  cardccrd 9358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6193  df-on 6194  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-card 9362
This theorem is referenced by:  cardiun  9405  carduniima  9516
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