Theorem rankunb 9608

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‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))

Proof of Theorem rankunb

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unwf 9568	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) ↔ (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
2		rankval3b 9584	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → (rank‘(𝐴 ∪ 𝐵)) = ∩ {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦})
3	1, 2	sylbi 216	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘(𝐴 ∪ 𝐵)) = ∩ {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦})
4	3	eleq2d 2824	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ 𝑥 ∈ ∩ {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦}))
5		vex 3436	. . . . . 6 ⊢ 𝑥 ∈ V
6	5	elintrab 4891	. . . . 5 ⊢ (𝑥 ∈ ∩ {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦} ↔ ∀𝑦 ∈ On (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦 → 𝑥 ∈ 𝑦))
7	4, 6	bitrdi 287	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ ∀𝑦 ∈ On (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦 → 𝑥 ∈ 𝑦)))
8		elun 4083	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
9		rankelb 9582	. . . . . . . . 9 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
10		elun1 4110	. . . . . . . . 9 ⊢ ((rank‘𝑥) ∈ (rank‘𝐴) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))
11	9, 10	syl6 35	. . . . . . . 8 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
12		rankelb 9582	. . . . . . . . 9 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐵 → (rank‘𝑥) ∈ (rank‘𝐵)))
13		elun2 4111	. . . . . . . . 9 ⊢ ((rank‘𝑥) ∈ (rank‘𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))
14	12, 13	syl6 35	. . . . . . . 8 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐵 → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
15	11, 14	jaao 952	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
16	8, 15	syl5bi 241	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝑥 ∈ (𝐴 ∪ 𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
17	16	ralrimiv 3102	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))
18		rankon 9553	. . . . . . 7 ⊢ (rank‘𝐴) ∈ On
19		rankon 9553	. . . . . . 7 ⊢ (rank‘𝐵) ∈ On
20	18, 19	onun2i 6382	. . . . . 6 ⊢ ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ On
21		eleq2 2827	. . . . . . . . 9 ⊢ (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → ((rank‘𝑥) ∈ 𝑦 ↔ (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
22	21	ralbidv 3112	. . . . . . . 8 ⊢ (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
23		eleq2 2827	. . . . . . . 8 ⊢ (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
24	22, 23	imbi12d 345	. . . . . . 7 ⊢ (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → ((∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦 → 𝑥 ∈ 𝑦) ↔ (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))))
25	24	rspcv 3557	. . . . . 6 ⊢ (((rank‘𝐴) ∪ (rank‘𝐵)) ∈ On → (∀𝑦 ∈ On (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦 → 𝑥 ∈ 𝑦) → (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))))
26	20, 25	ax-mp 5	. . . . 5 ⊢ (∀𝑦 ∈ On (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦 → 𝑥 ∈ 𝑦) → (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
27	17, 26	syl5com 31	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (∀𝑦 ∈ On (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦 → 𝑥 ∈ 𝑦) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
28	7, 27	sylbid 239	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴 ∪ 𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
29	28	ssrdv 3927	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘(𝐴 ∪ 𝐵)) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵)))
30		ssun1 4106	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
31		rankssb 9606	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → (𝐴 ⊆ (𝐴 ∪ 𝐵) → (rank‘𝐴) ⊆ (rank‘(𝐴 ∪ 𝐵))))
32	30, 31	mpi 20	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ⊆ (rank‘(𝐴 ∪ 𝐵)))
33		ssun2 4107	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
34		rankssb 9606	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → (𝐵 ⊆ (𝐴 ∪ 𝐵) → (rank‘𝐵) ⊆ (rank‘(𝐴 ∪ 𝐵))))
35	33, 34	mpi 20	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → (rank‘𝐵) ⊆ (rank‘(𝐴 ∪ 𝐵)))
36	32, 35	unssd 4120	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → ((rank‘𝐴) ∪ (rank‘𝐵)) ⊆ (rank‘(𝐴 ∪ 𝐵)))
37	1, 36	sylbi 216	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → ((rank‘𝐴) ∪ (rank‘𝐵)) ⊆ (rank‘(𝐴 ∪ 𝐵)))
38	29, 37	eqssd 3938	1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068   ∪ cun 3885   ⊆ wss 3887  ∪ cuni 4839  ∩ cint 4879   “ cima 5592  Oncon0 6266  ‘cfv 6433  𝑅₁cr1 9520  rankcrnk 9521

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-r1 9522  df-rank 9523

This theorem is referenced by:  rankprb  9609  rankopb  9610  rankun  9614  rankaltopb  34281