MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  carduni Structured version   Visualization version   GIF version

Theorem carduni 9978
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 (𝐴 ∈ 𝑉 β†’ (βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯ β†’ (cardβ€˜βˆͺ 𝐴) = βˆͺ 𝐴))
Distinct variable group:   π‘₯,𝐴
Allowed substitution hint:   𝑉(π‘₯)

Proof of Theorem carduni
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssonuni 7769 . . . . 5 (𝐴 ∈ 𝑉 β†’ (𝐴 βŠ† On β†’ βˆͺ 𝐴 ∈ On))
2 fveq2 6891 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ (cardβ€˜π‘₯) = (cardβ€˜π‘¦))
3 id 22 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ π‘₯ = 𝑦)
42, 3eqeq12d 2748 . . . . . . . 8 (π‘₯ = 𝑦 β†’ ((cardβ€˜π‘₯) = π‘₯ ↔ (cardβ€˜π‘¦) = 𝑦))
54rspcv 3608 . . . . . . 7 (𝑦 ∈ 𝐴 β†’ (βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯ β†’ (cardβ€˜π‘¦) = 𝑦))
6 cardon 9941 . . . . . . . 8 (cardβ€˜π‘¦) ∈ On
7 eleq1 2821 . . . . . . . 8 ((cardβ€˜π‘¦) = 𝑦 β†’ ((cardβ€˜π‘¦) ∈ On ↔ 𝑦 ∈ On))
86, 7mpbii 232 . . . . . . 7 ((cardβ€˜π‘¦) = 𝑦 β†’ 𝑦 ∈ On)
95, 8syl6com 37 . . . . . 6 (βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯ β†’ (𝑦 ∈ 𝐴 β†’ 𝑦 ∈ On))
109ssrdv 3988 . . . . 5 (βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯ β†’ 𝐴 βŠ† On)
111, 10impel 506 . . . 4 ((𝐴 ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯) β†’ βˆͺ 𝐴 ∈ On)
12 cardonle 9954 . . . 4 (βˆͺ 𝐴 ∈ On β†’ (cardβ€˜βˆͺ 𝐴) βŠ† βˆͺ 𝐴)
1311, 12syl 17 . . 3 ((𝐴 ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯) β†’ (cardβ€˜βˆͺ 𝐴) βŠ† βˆͺ 𝐴)
14 cardon 9941 . . . . 5 (cardβ€˜βˆͺ 𝐴) ∈ On
1514onirri 6477 . . . 4 Β¬ (cardβ€˜βˆͺ 𝐴) ∈ (cardβ€˜βˆͺ 𝐴)
16 eluni 4911 . . . . . . . 8 ((cardβ€˜βˆͺ 𝐴) ∈ βˆͺ 𝐴 ↔ βˆƒπ‘¦((cardβ€˜βˆͺ 𝐴) ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
17 elssuni 4941 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ 𝐴 β†’ 𝑦 βŠ† βˆͺ 𝐴)
18 ssdomg 8998 . . . . . . . . . . . . . . . . . . 19 (βˆͺ 𝐴 ∈ On β†’ (𝑦 βŠ† βˆͺ 𝐴 β†’ 𝑦 β‰Ό βˆͺ 𝐴))
1918adantl 482 . . . . . . . . . . . . . . . . . 18 (((cardβ€˜π‘¦) = 𝑦 ∧ βˆͺ 𝐴 ∈ On) β†’ (𝑦 βŠ† βˆͺ 𝐴 β†’ 𝑦 β‰Ό βˆͺ 𝐴))
2017, 19syl5 34 . . . . . . . . . . . . . . . . 17 (((cardβ€˜π‘¦) = 𝑦 ∧ βˆͺ 𝐴 ∈ On) β†’ (𝑦 ∈ 𝐴 β†’ 𝑦 β‰Ό βˆͺ 𝐴))
21 id 22 . . . . . . . . . . . . . . . . . . 19 ((cardβ€˜π‘¦) = 𝑦 β†’ (cardβ€˜π‘¦) = 𝑦)
22 onenon 9946 . . . . . . . . . . . . . . . . . . . 20 ((cardβ€˜π‘¦) ∈ On β†’ (cardβ€˜π‘¦) ∈ dom card)
236, 22ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (cardβ€˜π‘¦) ∈ dom card
2421, 23eqeltrrdi 2842 . . . . . . . . . . . . . . . . . 18 ((cardβ€˜π‘¦) = 𝑦 β†’ 𝑦 ∈ dom card)
25 onenon 9946 . . . . . . . . . . . . . . . . . 18 (βˆͺ 𝐴 ∈ On β†’ βˆͺ 𝐴 ∈ dom card)
26 carddom2 9974 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ dom card ∧ βˆͺ 𝐴 ∈ dom card) β†’ ((cardβ€˜π‘¦) βŠ† (cardβ€˜βˆͺ 𝐴) ↔ 𝑦 β‰Ό βˆͺ 𝐴))
2724, 25, 26syl2an 596 . . . . . . . . . . . . . . . . 17 (((cardβ€˜π‘¦) = 𝑦 ∧ βˆͺ 𝐴 ∈ On) β†’ ((cardβ€˜π‘¦) βŠ† (cardβ€˜βˆͺ 𝐴) ↔ 𝑦 β‰Ό βˆͺ 𝐴))
2820, 27sylibrd 258 . . . . . . . . . . . . . . . 16 (((cardβ€˜π‘¦) = 𝑦 ∧ βˆͺ 𝐴 ∈ On) β†’ (𝑦 ∈ 𝐴 β†’ (cardβ€˜π‘¦) βŠ† (cardβ€˜βˆͺ 𝐴)))
29 sseq1 4007 . . . . . . . . . . . . . . . . 17 ((cardβ€˜π‘¦) = 𝑦 β†’ ((cardβ€˜π‘¦) βŠ† (cardβ€˜βˆͺ 𝐴) ↔ 𝑦 βŠ† (cardβ€˜βˆͺ 𝐴)))
3029adantr 481 . . . . . . . . . . . . . . . 16 (((cardβ€˜π‘¦) = 𝑦 ∧ βˆͺ 𝐴 ∈ On) β†’ ((cardβ€˜π‘¦) βŠ† (cardβ€˜βˆͺ 𝐴) ↔ 𝑦 βŠ† (cardβ€˜βˆͺ 𝐴)))
3128, 30sylibd 238 . . . . . . . . . . . . . . 15 (((cardβ€˜π‘¦) = 𝑦 ∧ βˆͺ 𝐴 ∈ On) β†’ (𝑦 ∈ 𝐴 β†’ 𝑦 βŠ† (cardβ€˜βˆͺ 𝐴)))
32 ssel 3975 . . . . . . . . . . . . . . 15 (𝑦 βŠ† (cardβ€˜βˆͺ 𝐴) β†’ ((cardβ€˜βˆͺ 𝐴) ∈ 𝑦 β†’ (cardβ€˜βˆͺ 𝐴) ∈ (cardβ€˜βˆͺ 𝐴)))
3331, 32syl6 35 . . . . . . . . . . . . . 14 (((cardβ€˜π‘¦) = 𝑦 ∧ βˆͺ 𝐴 ∈ On) β†’ (𝑦 ∈ 𝐴 β†’ ((cardβ€˜βˆͺ 𝐴) ∈ 𝑦 β†’ (cardβ€˜βˆͺ 𝐴) ∈ (cardβ€˜βˆͺ 𝐴))))
3433ex 413 . . . . . . . . . . . . 13 ((cardβ€˜π‘¦) = 𝑦 β†’ (βˆͺ 𝐴 ∈ On β†’ (𝑦 ∈ 𝐴 β†’ ((cardβ€˜βˆͺ 𝐴) ∈ 𝑦 β†’ (cardβ€˜βˆͺ 𝐴) ∈ (cardβ€˜βˆͺ 𝐴)))))
3534com3r 87 . . . . . . . . . . . 12 (𝑦 ∈ 𝐴 β†’ ((cardβ€˜π‘¦) = 𝑦 β†’ (βˆͺ 𝐴 ∈ On β†’ ((cardβ€˜βˆͺ 𝐴) ∈ 𝑦 β†’ (cardβ€˜βˆͺ 𝐴) ∈ (cardβ€˜βˆͺ 𝐴)))))
365, 35syld 47 . . . . . . . . . . 11 (𝑦 ∈ 𝐴 β†’ (βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯ β†’ (βˆͺ 𝐴 ∈ On β†’ ((cardβ€˜βˆͺ 𝐴) ∈ 𝑦 β†’ (cardβ€˜βˆͺ 𝐴) ∈ (cardβ€˜βˆͺ 𝐴)))))
3736com4r 94 . . . . . . . . . 10 ((cardβ€˜βˆͺ 𝐴) ∈ 𝑦 β†’ (𝑦 ∈ 𝐴 β†’ (βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯ β†’ (βˆͺ 𝐴 ∈ On β†’ (cardβ€˜βˆͺ 𝐴) ∈ (cardβ€˜βˆͺ 𝐴)))))
3837imp 407 . . . . . . . . 9 (((cardβ€˜βˆͺ 𝐴) ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) β†’ (βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯ β†’ (βˆͺ 𝐴 ∈ On β†’ (cardβ€˜βˆͺ 𝐴) ∈ (cardβ€˜βˆͺ 𝐴))))
3938exlimiv 1933 . . . . . . . 8 (βˆƒπ‘¦((cardβ€˜βˆͺ 𝐴) ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) β†’ (βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯ β†’ (βˆͺ 𝐴 ∈ On β†’ (cardβ€˜βˆͺ 𝐴) ∈ (cardβ€˜βˆͺ 𝐴))))
4016, 39sylbi 216 . . . . . . 7 ((cardβ€˜βˆͺ 𝐴) ∈ βˆͺ 𝐴 β†’ (βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯ β†’ (βˆͺ 𝐴 ∈ On β†’ (cardβ€˜βˆͺ 𝐴) ∈ (cardβ€˜βˆͺ 𝐴))))
4140com13 88 . . . . . 6 (βˆͺ 𝐴 ∈ On β†’ (βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯ β†’ ((cardβ€˜βˆͺ 𝐴) ∈ βˆͺ 𝐴 β†’ (cardβ€˜βˆͺ 𝐴) ∈ (cardβ€˜βˆͺ 𝐴))))
4241imp 407 . . . . 5 ((βˆͺ 𝐴 ∈ On ∧ βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯) β†’ ((cardβ€˜βˆͺ 𝐴) ∈ βˆͺ 𝐴 β†’ (cardβ€˜βˆͺ 𝐴) ∈ (cardβ€˜βˆͺ 𝐴)))
4311, 42sylancom 588 . . . 4 ((𝐴 ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯) β†’ ((cardβ€˜βˆͺ 𝐴) ∈ βˆͺ 𝐴 β†’ (cardβ€˜βˆͺ 𝐴) ∈ (cardβ€˜βˆͺ 𝐴)))
4415, 43mtoi 198 . . 3 ((𝐴 ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯) β†’ Β¬ (cardβ€˜βˆͺ 𝐴) ∈ βˆͺ 𝐴)
4514onordi 6475 . . . 4 Ord (cardβ€˜βˆͺ 𝐴)
46 eloni 6374 . . . . 5 (βˆͺ 𝐴 ∈ On β†’ Ord βˆͺ 𝐴)
4711, 46syl 17 . . . 4 ((𝐴 ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯) β†’ Ord βˆͺ 𝐴)
48 ordtri4 6401 . . . 4 ((Ord (cardβ€˜βˆͺ 𝐴) ∧ Ord βˆͺ 𝐴) β†’ ((cardβ€˜βˆͺ 𝐴) = βˆͺ 𝐴 ↔ ((cardβ€˜βˆͺ 𝐴) βŠ† βˆͺ 𝐴 ∧ Β¬ (cardβ€˜βˆͺ 𝐴) ∈ βˆͺ 𝐴)))
4945, 47, 48sylancr 587 . . 3 ((𝐴 ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯) β†’ ((cardβ€˜βˆͺ 𝐴) = βˆͺ 𝐴 ↔ ((cardβ€˜βˆͺ 𝐴) βŠ† βˆͺ 𝐴 ∧ Β¬ (cardβ€˜βˆͺ 𝐴) ∈ βˆͺ 𝐴)))
5013, 44, 49mpbir2and 711 . 2 ((𝐴 ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯) β†’ (cardβ€˜βˆͺ 𝐴) = βˆͺ 𝐴)
5150ex 413 1 (𝐴 ∈ 𝑉 β†’ (βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯ β†’ (cardβ€˜βˆͺ 𝐴) = βˆͺ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  βˆ€wral 3061   βŠ† wss 3948  βˆͺ cuni 4908   class class class wbr 5148  dom cdm 5676  Ord word 6363  Oncon0 6364  β€˜cfv 6543   β‰Ό cdom 8939  cardccrd 9932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  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-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-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-ord 6367  df-on 6368  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-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-card 9936
This theorem is referenced by:  cardiun  9979  carduniima  10093
  Copyright terms: Public domain W3C validator