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Theorem rankunb 9890
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 9850 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) ↔ (𝐴𝐵) ∈ (𝑅1 “ On))
2 rankval3b 9866 . . . . . . 7 ((𝐴𝐵) ∈ (𝑅1 “ On) → (rank‘(𝐴𝐵)) = {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦})
31, 2sylbi 217 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘(𝐴𝐵)) = {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦})
43eleq2d 2827 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴𝐵)) ↔ 𝑥 {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦}))
5 vex 3484 . . . . . 6 𝑥 ∈ V
65elintrab 4960 . . . . 5 (𝑥 {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦} ↔ ∀𝑦 ∈ On (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦𝑥𝑦))
74, 6bitrdi 287 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴𝐵)) ↔ ∀𝑦 ∈ On (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦𝑥𝑦)))
8 elun 4153 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
9 rankelb 9864 . . . . . . . . 9 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
10 elun1 4182 . . . . . . . . 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 9864 . . . . . . . . 9 (𝐵 (𝑅1 “ On) → (𝑥𝐵 → (rank‘𝑥) ∈ (rank‘𝐵)))
13 elun2 4183 . . . . . . . . 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 957 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ((𝑥𝐴𝑥𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
168, 15biimtrid 242 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (𝐴𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
1716ralrimiv 3145 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))
18 rankon 9835 . . . . . . 7 (rank‘𝐴) ∈ On
19 rankon 9835 . . . . . . 7 (rank‘𝐵) ∈ On
2018, 19onun2i 6506 . . . . . 6 ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ On
21 eleq2 2830 . . . . . . . . 9 (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → ((rank‘𝑥) ∈ 𝑦 ↔ (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
2221ralbidv 3178 . . . . . . . 8 (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
23 eleq2 2830 . . . . . . . 8 (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → (𝑥𝑦𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
2422, 23imbi12d 344 . . . . . . 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 3618 . . . . . 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 240 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
2928ssrdv 3989 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘(𝐴𝐵)) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵)))
30 ssun1 4178 . . . . 5 𝐴 ⊆ (𝐴𝐵)
31 rankssb 9888 . . . . 5 ((𝐴𝐵) ∈ (𝑅1 “ On) → (𝐴 ⊆ (𝐴𝐵) → (rank‘𝐴) ⊆ (rank‘(𝐴𝐵))))
3230, 31mpi 20 . . . 4 ((𝐴𝐵) ∈ (𝑅1 “ On) → (rank‘𝐴) ⊆ (rank‘(𝐴𝐵)))
33 ssun2 4179 . . . . 5 𝐵 ⊆ (𝐴𝐵)
34 rankssb 9888 . . . . 5 ((𝐴𝐵) ∈ (𝑅1 “ On) → (𝐵 ⊆ (𝐴𝐵) → (rank‘𝐵) ⊆ (rank‘(𝐴𝐵))))
3533, 34mpi 20 . . . 4 ((𝐴𝐵) ∈ (𝑅1 “ On) → (rank‘𝐵) ⊆ (rank‘(𝐴𝐵)))
3632, 35unssd 4192 . . 3 ((𝐴𝐵) ∈ (𝑅1 “ On) → ((rank‘𝐴) ∪ (rank‘𝐵)) ⊆ (rank‘(𝐴𝐵)))
371, 36sylbi 217 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ((rank‘𝐴) ∪ (rank‘𝐵)) ⊆ (rank‘(𝐴𝐵)))
3829, 37eqssd 4001 1 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848   = wceq 1540  wcel 2108  wral 3061  {crab 3436  cun 3949  wss 3951   cuni 4907   cint 4946  cima 5688  Oncon0 6384  cfv 6561  𝑅1cr1 9802  rankcrnk 9803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-r1 9804  df-rank 9805
This theorem is referenced by:  rankprb  9891  rankopb  9892  rankun  9896  rankaltopb  35980
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