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

Theorem rankunb 9267
Description: The rank of the union of two sets. Theorem 15.17(iii) of [Monk1] p. 112. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
rankunb ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))

Proof of Theorem rankunb
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unwf 9227 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) ↔ (𝐴𝐵) ∈ (𝑅1 “ On))
2 rankval3b 9243 . . . . . . 7 ((𝐴𝐵) ∈ (𝑅1 “ On) → (rank‘(𝐴𝐵)) = {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦})
31, 2sylbi 218 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘(𝐴𝐵)) = {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦})
43eleq2d 2895 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴𝐵)) ↔ 𝑥 {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦}))
5 vex 3495 . . . . . 6 𝑥 ∈ V
65elintrab 4879 . . . . 5 (𝑥 {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦} ↔ ∀𝑦 ∈ On (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦𝑥𝑦))
74, 6syl6bb 288 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴𝐵)) ↔ ∀𝑦 ∈ On (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦𝑥𝑦)))
8 elun 4122 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
9 rankelb 9241 . . . . . . . . 9 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
10 elun1 4149 . . . . . . . . 9 ((rank‘𝑥) ∈ (rank‘𝐴) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))
119, 10syl6 35 . . . . . . . 8 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
12 rankelb 9241 . . . . . . . . 9 (𝐵 (𝑅1 “ On) → (𝑥𝐵 → (rank‘𝑥) ∈ (rank‘𝐵)))
13 elun2 4150 . . . . . . . . 9 ((rank‘𝑥) ∈ (rank‘𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))
1412, 13syl6 35 . . . . . . . 8 (𝐵 (𝑅1 “ On) → (𝑥𝐵 → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
1511, 14jaao 948 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ((𝑥𝐴𝑥𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
168, 15syl5bi 243 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (𝐴𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
1716ralrimiv 3178 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))
18 rankon 9212 . . . . . . 7 (rank‘𝐴) ∈ On
19 rankon 9212 . . . . . . 7 (rank‘𝐵) ∈ On
2018, 19onun2i 6299 . . . . . 6 ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ On
21 eleq2 2898 . . . . . . . . 9 (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → ((rank‘𝑥) ∈ 𝑦 ↔ (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
2221ralbidv 3194 . . . . . . . 8 (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
23 eleq2 2898 . . . . . . . 8 (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → (𝑥𝑦𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
2422, 23imbi12d 346 . . . . . . 7 (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → ((∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦𝑥𝑦) ↔ (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))))
2524rspcv 3615 . . . . . 6 (((rank‘𝐴) ∪ (rank‘𝐵)) ∈ On → (∀𝑦 ∈ On (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦𝑥𝑦) → (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))))
2620, 25ax-mp 5 . . . . 5 (∀𝑦 ∈ On (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦𝑥𝑦) → (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
2717, 26syl5com 31 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (∀𝑦 ∈ On (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦𝑥𝑦) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
287, 27sylbid 241 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
2928ssrdv 3970 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘(𝐴𝐵)) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵)))
30 ssun1 4145 . . . . 5 𝐴 ⊆ (𝐴𝐵)
31 rankssb 9265 . . . . 5 ((𝐴𝐵) ∈ (𝑅1 “ On) → (𝐴 ⊆ (𝐴𝐵) → (rank‘𝐴) ⊆ (rank‘(𝐴𝐵))))
3230, 31mpi 20 . . . 4 ((𝐴𝐵) ∈ (𝑅1 “ On) → (rank‘𝐴) ⊆ (rank‘(𝐴𝐵)))
33 ssun2 4146 . . . . 5 𝐵 ⊆ (𝐴𝐵)
34 rankssb 9265 . . . . 5 ((𝐴𝐵) ∈ (𝑅1 “ On) → (𝐵 ⊆ (𝐴𝐵) → (rank‘𝐵) ⊆ (rank‘(𝐴𝐵))))
3533, 34mpi 20 . . . 4 ((𝐴𝐵) ∈ (𝑅1 “ On) → (rank‘𝐵) ⊆ (rank‘(𝐴𝐵)))
3632, 35unssd 4159 . . 3 ((𝐴𝐵) ∈ (𝑅1 “ On) → ((rank‘𝐴) ∪ (rank‘𝐵)) ⊆ (rank‘(𝐴𝐵)))
371, 36sylbi 218 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ((rank‘𝐴) ∪ (rank‘𝐵)) ⊆ (rank‘(𝐴𝐵)))
3829, 37eqssd 3981 1 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 841   = wceq 1528  wcel 2105  wral 3135  {crab 3139  cun 3931  wss 3933   cuni 4830   cint 4867  cima 5551  Oncon0 6184  cfv 6348  𝑅1cr1 9179  rankcrnk 9180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-r1 9181  df-rank 9182
This theorem is referenced by:  rankprb  9268  rankopb  9269  rankun  9273  rankaltopb  33337
  Copyright terms: Public domain W3C validator