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Theorem rankunb 8665
 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 8625 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) ↔ (𝐴𝐵) ∈ (𝑅1 “ On))
2 rankval3b 8641 . . . . . . 7 ((𝐴𝐵) ∈ (𝑅1 “ On) → (rank‘(𝐴𝐵)) = {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦})
31, 2sylbi 207 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘(𝐴𝐵)) = {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦})
43eleq2d 2684 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴𝐵)) ↔ 𝑥 {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦}))
5 vex 3192 . . . . . 6 𝑥 ∈ V
65elintrab 4458 . . . . 5 (𝑥 {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦} ↔ ∀𝑦 ∈ On (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦𝑥𝑦))
74, 6syl6bb 276 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴𝐵)) ↔ ∀𝑦 ∈ On (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦𝑥𝑦)))
8 elun 3736 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
9 rankelb 8639 . . . . . . . . 9 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
10 elun1 3763 . . . . . . . . 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 8639 . . . . . . . . 9 (𝐵 (𝑅1 “ On) → (𝑥𝐵 → (rank‘𝑥) ∈ (rank‘𝐵)))
13 elun2 3764 . . . . . . . . 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 531 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ((𝑥𝐴𝑥𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
168, 15syl5bi 232 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (𝐴𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
1716ralrimiv 2960 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))
18 rankon 8610 . . . . . . 7 (rank‘𝐴) ∈ On
19 rankon 8610 . . . . . . 7 (rank‘𝐵) ∈ On
2018, 19onun2i 5807 . . . . . 6 ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ On
21 eleq2 2687 . . . . . . . . 9 (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → ((rank‘𝑥) ∈ 𝑦 ↔ (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
2221ralbidv 2981 . . . . . . . 8 (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
23 eleq2 2687 . . . . . . . 8 (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → (𝑥𝑦𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
2422, 23imbi12d 334 . . . . . . 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 3294 . . . . . 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 230 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
2928ssrdv 3593 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘(𝐴𝐵)) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵)))
30 ssun1 3759 . . . . 5 𝐴 ⊆ (𝐴𝐵)
31 rankssb 8663 . . . . 5 ((𝐴𝐵) ∈ (𝑅1 “ On) → (𝐴 ⊆ (𝐴𝐵) → (rank‘𝐴) ⊆ (rank‘(𝐴𝐵))))
3230, 31mpi 20 . . . 4 ((𝐴𝐵) ∈ (𝑅1 “ On) → (rank‘𝐴) ⊆ (rank‘(𝐴𝐵)))
33 ssun2 3760 . . . . 5 𝐵 ⊆ (𝐴𝐵)
34 rankssb 8663 . . . . 5 ((𝐴𝐵) ∈ (𝑅1 “ On) → (𝐵 ⊆ (𝐴𝐵) → (rank‘𝐵) ⊆ (rank‘(𝐴𝐵))))
3533, 34mpi 20 . . . 4 ((𝐴𝐵) ∈ (𝑅1 “ On) → (rank‘𝐵) ⊆ (rank‘(𝐴𝐵)))
3632, 35unssd 3772 . . 3 ((𝐴𝐵) ∈ (𝑅1 “ On) → ((rank‘𝐴) ∪ (rank‘𝐵)) ⊆ (rank‘(𝐴𝐵)))
371, 36sylbi 207 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ((rank‘𝐴) ∪ (rank‘𝐵)) ⊆ (rank‘(𝐴𝐵)))
3829, 37eqssd 3604 1 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907  {crab 2911   ∪ cun 3557   ⊆ wss 3559  ∪ cuni 4407  ∩ cint 4445   “ cima 5082  Oncon0 5687  ‘cfv 5852  𝑅1cr1 8577  rankcrnk 8578 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-r1 8579  df-rank 8580 This theorem is referenced by:  rankprb  8666  rankopb  8667  rankun  8671  rankaltopb  31763
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