Theorem rankunb 9888

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 9848	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) ↔ (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
2		rankval3b 9864	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → (rank‘(𝐴 ∪ 𝐵)) = ∩ {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦})
3	1, 2	sylbi 217	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘(𝐴 ∪ 𝐵)) = ∩ {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦})
4	3	eleq2d 2825	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ 𝑥 ∈ ∩ {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦}))
5		vex 3482	. . . . . 6 ⊢ 𝑥 ∈ V
6	5	elintrab 4965	. . . . 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 4163	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
9		rankelb 9862	. . . . . . . . 9 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
10		elun1 4192	. . . . . . . . 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 9862	. . . . . . . . 9 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐵 → (rank‘𝑥) ∈ (rank‘𝐵)))
13		elun2 4193	. . . . . . . . 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 956	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
16	8, 15	biimtrid 242	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝑥 ∈ (𝐴 ∪ 𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
17	16	ralrimiv 3143	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))
18		rankon 9833	. . . . . . 7 ⊢ (rank‘𝐴) ∈ On
19		rankon 9833	. . . . . . 7 ⊢ (rank‘𝐵) ∈ On
20	18, 19	onun2i 6508	. . . . . 6 ⊢ ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ On
21		eleq2 2828	. . . . . . . . 9 ⊢ (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → ((rank‘𝑥) ∈ 𝑦 ↔ (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
22	21	ralbidv 3176	. . . . . . . 8 ⊢ (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
23		eleq2 2828	. . . . . . . 8 ⊢ (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
24	22, 23	imbi12d 344	. . . . . . 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 3618	. . . . . 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 240	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴 ∪ 𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
29	28	ssrdv 4001	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘(𝐴 ∪ 𝐵)) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵)))
30		ssun1 4188	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
31		rankssb 9886	. . . . 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 4189	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
34		rankssb 9886	. . . . 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 4202	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → ((rank‘𝐴) ∪ (rank‘𝐵)) ⊆ (rank‘(𝐴 ∪ 𝐵)))
37	1, 36	sylbi 217	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → ((rank‘𝐴) ∪ (rank‘𝐵)) ⊆ (rank‘(𝐴 ∪ 𝐵)))
38	29, 37	eqssd 4013	1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433   ∪ cun 3961   ⊆ wss 3963  ∪ cuni 4912  ∩ cint 4951   “ cima 5692  Oncon0 6386  ‘cfv 6563  𝑅₁cr1 9800  rankcrnk 9801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-r1 9802  df-rank 9803

This theorem is referenced by:  rankprb  9889  rankopb  9890  rankun  9894  rankaltopb  35961