Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ficardun2 Structured version   Visualization version   GIF version

Theorem ficardun2 9602
 Description: The cardinality of the union of finite sets is at most the ordinal sum of their cardinalities. (Contributed by Mario Carneiro, 5-Feb-2013.)
Assertion
Ref Expression
ficardun2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴𝐵)) ⊆ ((card‘𝐴) +o (card‘𝐵)))

Proof of Theorem ficardun2
StepHypRef Expression
1 undjudom 9570 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≼ (𝐴𝐵))
2 finnum 9353 . . . . 5 (𝐴 ∈ Fin → 𝐴 ∈ dom card)
3 finnum 9353 . . . . 5 (𝐵 ∈ Fin → 𝐵 ∈ dom card)
4 cardadju 9597 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))
52, 3, 4syl2an 598 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))
6 domentr 8543 . . . 4 (((𝐴𝐵) ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≈ ((card‘𝐴) +o (card‘𝐵))) → (𝐴𝐵) ≼ ((card‘𝐴) +o (card‘𝐵)))
71, 5, 6syl2anc 587 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≼ ((card‘𝐴) +o (card‘𝐵)))
8 unfi 8761 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin)
9 finnum 9353 . . . . 5 ((𝐴𝐵) ∈ Fin → (𝐴𝐵) ∈ dom card)
108, 9syl 17 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ dom card)
11 ficardom 9366 . . . . . 6 (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
12 ficardom 9366 . . . . . 6 (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
13 nnacl 8212 . . . . . 6 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +o (card‘𝐵)) ∈ ω)
1411, 12, 13syl2an 598 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ∈ ω)
15 nnon 7561 . . . . 5 (((card‘𝐴) +o (card‘𝐵)) ∈ ω → ((card‘𝐴) +o (card‘𝐵)) ∈ On)
16 onenon 9354 . . . . 5 (((card‘𝐴) +o (card‘𝐵)) ∈ On → ((card‘𝐴) +o (card‘𝐵)) ∈ dom card)
1714, 15, 163syl 18 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ∈ dom card)
18 carddom2 9382 . . . 4 (((𝐴𝐵) ∈ dom card ∧ ((card‘𝐴) +o (card‘𝐵)) ∈ dom card) → ((card‘(𝐴𝐵)) ⊆ (card‘((card‘𝐴) +o (card‘𝐵))) ↔ (𝐴𝐵) ≼ ((card‘𝐴) +o (card‘𝐵))))
1910, 17, 18syl2anc 587 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘(𝐴𝐵)) ⊆ (card‘((card‘𝐴) +o (card‘𝐵))) ↔ (𝐴𝐵) ≼ ((card‘𝐴) +o (card‘𝐵))))
207, 19mpbird 260 . 2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴𝐵)) ⊆ (card‘((card‘𝐴) +o (card‘𝐵))))
21 cardnn 9368 . . 3 (((card‘𝐴) +o (card‘𝐵)) ∈ ω → (card‘((card‘𝐴) +o (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
2214, 21syl 17 . 2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘((card‘𝐴) +o (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
2320, 22sseqtrd 3983 1 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴𝐵)) ⊆ ((card‘𝐴) +o (card‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115   ∪ cun 3908   ⊆ wss 3910   class class class wbr 5039  dom cdm 5528  Oncon0 6164  ‘cfv 6328  (class class class)co 7130  ωcom 7555   +o coa 8074   ≈ cen 8481   ≼ cdom 8482  Fincfn 8484   ⊔ cdju 9303  cardccrd 9340 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-dju 9306  df-card 9344 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator