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

Theorem carduni 9922
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 7715 . . . . 5 (𝐴 ∈ 𝑉 β†’ (𝐴 βŠ† On β†’ βˆͺ 𝐴 ∈ On))
2 fveq2 6843 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ (cardβ€˜π‘₯) = (cardβ€˜π‘¦))
3 id 22 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ π‘₯ = 𝑦)
42, 3eqeq12d 2749 . . . . . . . 8 (π‘₯ = 𝑦 β†’ ((cardβ€˜π‘₯) = π‘₯ ↔ (cardβ€˜π‘¦) = 𝑦))
54rspcv 3576 . . . . . . 7 (𝑦 ∈ 𝐴 β†’ (βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯ β†’ (cardβ€˜π‘¦) = 𝑦))
6 cardon 9885 . . . . . . . 8 (cardβ€˜π‘¦) ∈ On
7 eleq1 2822 . . . . . . . 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 3951 . . . . 5 (βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯ β†’ 𝐴 βŠ† On)
111, 10impel 507 . . . 4 ((𝐴 ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯) β†’ βˆͺ 𝐴 ∈ On)
12 cardonle 9898 . . . 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 9885 . . . . 5 (cardβ€˜βˆͺ 𝐴) ∈ On
1514onirri 6431 . . . 4 Β¬ (cardβ€˜βˆͺ 𝐴) ∈ (cardβ€˜βˆͺ 𝐴)
16 eluni 4869 . . . . . . . 8 ((cardβ€˜βˆͺ 𝐴) ∈ βˆͺ 𝐴 ↔ βˆƒπ‘¦((cardβ€˜βˆͺ 𝐴) ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
17 elssuni 4899 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ 𝐴 β†’ 𝑦 βŠ† βˆͺ 𝐴)
18 ssdomg 8943 . . . . . . . . . . . . . . . . . . 19 (βˆͺ 𝐴 ∈ On β†’ (𝑦 βŠ† βˆͺ 𝐴 β†’ 𝑦 β‰Ό βˆͺ 𝐴))
1918adantl 483 . . . . . . . . . . . . . . . . . 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 9890 . . . . . . . . . . . . . . . . . . . 20 ((cardβ€˜π‘¦) ∈ On β†’ (cardβ€˜π‘¦) ∈ dom card)
236, 22ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (cardβ€˜π‘¦) ∈ dom card
2421, 23eqeltrrdi 2843 . . . . . . . . . . . . . . . . . 18 ((cardβ€˜π‘¦) = 𝑦 β†’ 𝑦 ∈ dom card)
25 onenon 9890 . . . . . . . . . . . . . . . . . 18 (βˆͺ 𝐴 ∈ On β†’ βˆͺ 𝐴 ∈ dom card)
26 carddom2 9918 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ dom card ∧ βˆͺ 𝐴 ∈ dom card) β†’ ((cardβ€˜π‘¦) βŠ† (cardβ€˜βˆͺ 𝐴) ↔ 𝑦 β‰Ό βˆͺ 𝐴))
2724, 25, 26syl2an 597 . . . . . . . . . . . . . . . . 17 (((cardβ€˜π‘¦) = 𝑦 ∧ βˆͺ 𝐴 ∈ On) β†’ ((cardβ€˜π‘¦) βŠ† (cardβ€˜βˆͺ 𝐴) ↔ 𝑦 β‰Ό βˆͺ 𝐴))
2820, 27sylibrd 259 . . . . . . . . . . . . . . . 16 (((cardβ€˜π‘¦) = 𝑦 ∧ βˆͺ 𝐴 ∈ On) β†’ (𝑦 ∈ 𝐴 β†’ (cardβ€˜π‘¦) βŠ† (cardβ€˜βˆͺ 𝐴)))
29 sseq1 3970 . . . . . . . . . . . . . . . . 17 ((cardβ€˜π‘¦) = 𝑦 β†’ ((cardβ€˜π‘¦) βŠ† (cardβ€˜βˆͺ 𝐴) ↔ 𝑦 βŠ† (cardβ€˜βˆͺ 𝐴)))
3029adantr 482 . . . . . . . . . . . . . . . 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 3938 . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . 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 408 . . . . . . . . 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 1934 . . . . . . . 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 408 . . . . 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 589 . . . 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 6429 . . . 4 Ord (cardβ€˜βˆͺ 𝐴)
46 eloni 6328 . . . . 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 6355 . . . 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 588 . . 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 712 . 2 ((𝐴 ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝐴 (cardβ€˜π‘₯) = π‘₯) β†’ (cardβ€˜βˆͺ 𝐴) = βˆͺ 𝐴)
5150ex 414 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 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆ€wral 3061   βŠ† wss 3911  βˆͺ cuni 4866   class class class wbr 5106  dom cdm 5634  Ord word 6317  Oncon0 6318  β€˜cfv 6497   β‰Ό cdom 8884  cardccrd 9876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-card 9880
This theorem is referenced by:  cardiun  9923  carduniima  10037
  Copyright terms: Public domain W3C validator