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Theorem carduni 9028
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 6348 . . . . . . . . . 10 (𝑥 = 𝑦 → (card‘𝑥) = (card‘𝑦))
2 id 22 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
31, 2eqeq12d 2789 . . . . . . . . 9 (𝑥 = 𝑦 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑦) = 𝑦))
43rspcv 3461 . . . . . . . 8 (𝑦𝐴 → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → (card‘𝑦) = 𝑦))
5 cardon 8991 . . . . . . . . 9 (card‘𝑦) ∈ On
6 eleq1 2841 . . . . . . . . 9 ((card‘𝑦) = 𝑦 → ((card‘𝑦) ∈ On ↔ 𝑦 ∈ On))
75, 6mpbii 224 . . . . . . . 8 ((card‘𝑦) = 𝑦𝑦 ∈ On)
84, 7syl6com 37 . . . . . . 7 (∀𝑥𝐴 (card‘𝑥) = 𝑥 → (𝑦𝐴𝑦 ∈ On))
98ssrdv 3764 . . . . . 6 (∀𝑥𝐴 (card‘𝑥) = 𝑥𝐴 ⊆ On)
10 ssonuni 7154 . . . . . 6 (𝐴𝑉 → (𝐴 ⊆ On → 𝐴 ∈ On))
119, 10syl5 34 . . . . 5 (𝐴𝑉 → (∀𝑥𝐴 (card‘𝑥) = 𝑥 𝐴 ∈ On))
1211imp 394 . . . 4 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → 𝐴 ∈ On)
13 cardonle 9004 . . . 4 ( 𝐴 ∈ On → (card‘ 𝐴) ⊆ 𝐴)
1412, 13syl 17 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → (card‘ 𝐴) ⊆ 𝐴)
15 cardon 8991 . . . . 5 (card‘ 𝐴) ∈ On
1615onirri 5988 . . . 4 ¬ (card‘ 𝐴) ∈ (card‘ 𝐴)
17 eluni 4588 . . . . . . . 8 ((card‘ 𝐴) ∈ 𝐴 ↔ ∃𝑦((card‘ 𝐴) ∈ 𝑦𝑦𝐴))
18 elssuni 4614 . . . . . . . . . . . . . . . . . 18 (𝑦𝐴𝑦 𝐴)
19 ssdomg 8176 . . . . . . . . . . . . . . . . . . 19 ( 𝐴 ∈ On → (𝑦 𝐴𝑦 𝐴))
2019adantl 468 . . . . . . . . . . . . . . . . . 18 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → (𝑦 𝐴𝑦 𝐴))
2118, 20syl5 34 . . . . . . . . . . . . . . . . 17 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → (𝑦𝐴𝑦 𝐴))
22 id 22 . . . . . . . . . . . . . . . . . . 19 ((card‘𝑦) = 𝑦 → (card‘𝑦) = 𝑦)
23 onenon 8996 . . . . . . . . . . . . . . . . . . . 20 ((card‘𝑦) ∈ On → (card‘𝑦) ∈ dom card)
245, 23ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (card‘𝑦) ∈ dom card
2522, 24syl6eqelr 2862 . . . . . . . . . . . . . . . . . 18 ((card‘𝑦) = 𝑦𝑦 ∈ dom card)
26 onenon 8996 . . . . . . . . . . . . . . . . . 18 ( 𝐴 ∈ On → 𝐴 ∈ dom card)
27 carddom2 9024 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘𝑦) ⊆ (card‘ 𝐴) ↔ 𝑦 𝐴))
2825, 26, 27syl2an 584 . . . . . . . . . . . . . . . . 17 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → ((card‘𝑦) ⊆ (card‘ 𝐴) ↔ 𝑦 𝐴))
2921, 28sylibrd 250 . . . . . . . . . . . . . . . 16 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → (𝑦𝐴 → (card‘𝑦) ⊆ (card‘ 𝐴)))
30 sseq1 3782 . . . . . . . . . . . . . . . . 17 ((card‘𝑦) = 𝑦 → ((card‘𝑦) ⊆ (card‘ 𝐴) ↔ 𝑦 ⊆ (card‘ 𝐴)))
3130adantr 467 . . . . . . . . . . . . . . . 16 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → ((card‘𝑦) ⊆ (card‘ 𝐴) ↔ 𝑦 ⊆ (card‘ 𝐴)))
3229, 31sylibd 230 . . . . . . . . . . . . . . 15 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → (𝑦𝐴𝑦 ⊆ (card‘ 𝐴)))
33 ssel 3752 . . . . . . . . . . . . . . 15 (𝑦 ⊆ (card‘ 𝐴) → ((card‘ 𝐴) ∈ 𝑦 → (card‘ 𝐴) ∈ (card‘ 𝐴)))
3432, 33syl6 35 . . . . . . . . . . . . . 14 (((card‘𝑦) = 𝑦 𝐴 ∈ On) → (𝑦𝐴 → ((card‘ 𝐴) ∈ 𝑦 → (card‘ 𝐴) ∈ (card‘ 𝐴))))
3534ex 398 . . . . . . . . . . . . 13 ((card‘𝑦) = 𝑦 → ( 𝐴 ∈ On → (𝑦𝐴 → ((card‘ 𝐴) ∈ 𝑦 → (card‘ 𝐴) ∈ (card‘ 𝐴)))))
3635com3r 87 . . . . . . . . . . . 12 (𝑦𝐴 → ((card‘𝑦) = 𝑦 → ( 𝐴 ∈ On → ((card‘ 𝐴) ∈ 𝑦 → (card‘ 𝐴) ∈ (card‘ 𝐴)))))
374, 36syld 47 . . . . . . . . . . 11 (𝑦𝐴 → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → ( 𝐴 ∈ On → ((card‘ 𝐴) ∈ 𝑦 → (card‘ 𝐴) ∈ (card‘ 𝐴)))))
3837com4r 94 . . . . . . . . . 10 ((card‘ 𝐴) ∈ 𝑦 → (𝑦𝐴 → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → ( 𝐴 ∈ On → (card‘ 𝐴) ∈ (card‘ 𝐴)))))
3938imp 394 . . . . . . . . 9 (((card‘ 𝐴) ∈ 𝑦𝑦𝐴) → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → ( 𝐴 ∈ On → (card‘ 𝐴) ∈ (card‘ 𝐴))))
4039exlimiv 2013 . . . . . . . 8 (∃𝑦((card‘ 𝐴) ∈ 𝑦𝑦𝐴) → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → ( 𝐴 ∈ On → (card‘ 𝐴) ∈ (card‘ 𝐴))))
4117, 40sylbi 208 . . . . . . 7 ((card‘ 𝐴) ∈ 𝐴 → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → ( 𝐴 ∈ On → (card‘ 𝐴) ∈ (card‘ 𝐴))))
4241com13 88 . . . . . 6 ( 𝐴 ∈ On → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → ((card‘ 𝐴) ∈ 𝐴 → (card‘ 𝐴) ∈ (card‘ 𝐴))))
4342imp 394 . . . . 5 (( 𝐴 ∈ On ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → ((card‘ 𝐴) ∈ 𝐴 → (card‘ 𝐴) ∈ (card‘ 𝐴)))
4412, 43sylancom 577 . . . 4 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → ((card‘ 𝐴) ∈ 𝐴 → (card‘ 𝐴) ∈ (card‘ 𝐴)))
4516, 44mtoi 190 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → ¬ (card‘ 𝐴) ∈ 𝐴)
4615onordi 5986 . . . 4 Ord (card‘ 𝐴)
47 eloni 5887 . . . . 5 ( 𝐴 ∈ On → Ord 𝐴)
4812, 47syl 17 . . . 4 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → Ord 𝐴)
49 ordtri4 5915 . . . 4 ((Ord (card‘ 𝐴) ∧ Ord 𝐴) → ((card‘ 𝐴) = 𝐴 ↔ ((card‘ 𝐴) ⊆ 𝐴 ∧ ¬ (card‘ 𝐴) ∈ 𝐴)))
5046, 48, 49sylancr 576 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → ((card‘ 𝐴) = 𝐴 ↔ ((card‘ 𝐴) ⊆ 𝐴 ∧ ¬ (card‘ 𝐴) ∈ 𝐴)))
5114, 45, 50mpbir2and 693 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 (card‘𝑥) = 𝑥) → (card‘ 𝐴) = 𝐴)
5251ex 398 1 (𝐴𝑉 → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → (card‘ 𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 383   = wceq 1634  wex 1855  wcel 2148  wral 3064  wss 3729   cuni 4585   class class class wbr 4797  dom cdm 5263  Ord word 5876  Oncon0 5877  cfv 6042  cdom 8128  cardccrd 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-sbc 3594  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-int 4623  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-ord 5880  df-on 5881  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-er 7917  df-en 8131  df-dom 8132  df-sdom 8133  df-card 8986
This theorem is referenced by:  cardiun  9029  carduniima  9140
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