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Theorem rankunb 9302
 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 9262 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) ↔ (𝐴𝐵) ∈ (𝑅1 “ On))
2 rankval3b 9278 . . . . . . 7 ((𝐴𝐵) ∈ (𝑅1 “ On) → (rank‘(𝐴𝐵)) = {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦})
31, 2sylbi 220 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘(𝐴𝐵)) = {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦})
43eleq2d 2838 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴𝐵)) ↔ 𝑥 {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦}))
5 vex 3414 . . . . . 6 𝑥 ∈ V
65elintrab 4848 . . . . 5 (𝑥 {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦} ↔ ∀𝑦 ∈ On (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦𝑥𝑦))
74, 6bitrdi 290 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴𝐵)) ↔ ∀𝑦 ∈ On (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦𝑥𝑦)))
8 elun 4055 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
9 rankelb 9276 . . . . . . . . 9 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
10 elun1 4082 . . . . . . . . 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 9276 . . . . . . . . 9 (𝐵 (𝑅1 “ On) → (𝑥𝐵 → (rank‘𝑥) ∈ (rank‘𝐵)))
13 elun2 4083 . . . . . . . . 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 953 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ((𝑥𝐴𝑥𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
168, 15syl5bi 245 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (𝐴𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
1716ralrimiv 3113 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))
18 rankon 9247 . . . . . . 7 (rank‘𝐴) ∈ On
19 rankon 9247 . . . . . . 7 (rank‘𝐵) ∈ On
2018, 19onun2i 6283 . . . . . 6 ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ On
21 eleq2 2841 . . . . . . . . 9 (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → ((rank‘𝑥) ∈ 𝑦 ↔ (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
2221ralbidv 3127 . . . . . . . 8 (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
23 eleq2 2841 . . . . . . . 8 (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → (𝑥𝑦𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
2422, 23imbi12d 349 . . . . . . 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 3537 . . . . . 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 243 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
2928ssrdv 3899 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘(𝐴𝐵)) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵)))
30 ssun1 4078 . . . . 5 𝐴 ⊆ (𝐴𝐵)
31 rankssb 9300 . . . . 5 ((𝐴𝐵) ∈ (𝑅1 “ On) → (𝐴 ⊆ (𝐴𝐵) → (rank‘𝐴) ⊆ (rank‘(𝐴𝐵))))
3230, 31mpi 20 . . . 4 ((𝐴𝐵) ∈ (𝑅1 “ On) → (rank‘𝐴) ⊆ (rank‘(𝐴𝐵)))
33 ssun2 4079 . . . . 5 𝐵 ⊆ (𝐴𝐵)
34 rankssb 9300 . . . . 5 ((𝐴𝐵) ∈ (𝑅1 “ On) → (𝐵 ⊆ (𝐴𝐵) → (rank‘𝐵) ⊆ (rank‘(𝐴𝐵))))
3533, 34mpi 20 . . . 4 ((𝐴𝐵) ∈ (𝑅1 “ On) → (rank‘𝐵) ⊆ (rank‘(𝐴𝐵)))
3632, 35unssd 4092 . . 3 ((𝐴𝐵) ∈ (𝑅1 “ On) → ((rank‘𝐴) ∪ (rank‘𝐵)) ⊆ (rank‘(𝐴𝐵)))
371, 36sylbi 220 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ((rank‘𝐴) ∪ (rank‘𝐵)) ⊆ (rank‘(𝐴𝐵)))
3829, 37eqssd 3910 1 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 845   = wceq 1539   ∈ wcel 2112  ∀wral 3071  {crab 3075   ∪ cun 3857   ⊆ wss 3859  ∪ cuni 4796  ∩ cint 4836   “ cima 5525  Oncon0 6167  ‘cfv 6333  𝑅1cr1 9214  rankcrnk 9215 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7457 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4419  df-pw 4494  df-sn 4521  df-pr 4523  df-tp 4525  df-op 4527  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6292  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-om 7578  df-wrecs 7955  df-recs 8016  df-rdg 8054  df-r1 9216  df-rank 9217 This theorem is referenced by:  rankprb  9303  rankopb  9304  rankun  9308  rankaltopb  33820
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