Theorem rankunb 9847

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 9807	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) ↔ (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
2		rankval3b 9823	. . . . . . 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 2817	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ 𝑥 ∈ ∩ {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦}))
5		vex 3476	. . . . . 6 ⊢ 𝑥 ∈ V
6	5	elintrab 4963	. . . . 5 ⊢ (𝑥 ∈ ∩ {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦} ↔ ∀𝑦 ∈ On (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦 → 𝑥 ∈ 𝑦))
7	4, 6	bitrdi 286	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ ∀𝑦 ∈ On (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦 → 𝑥 ∈ 𝑦)))
8		elun 4147	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
9		rankelb 9821	. . . . . . . . 9 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
10		elun1 4175	. . . . . . . . 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 9821	. . . . . . . . 9 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐵 → (rank‘𝑥) ∈ (rank‘𝐵)))
13		elun2 4176	. . . . . . . . 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 951	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
16	8, 15	biimtrid 241	. . . . . 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 9792	. . . . . . 7 ⊢ (rank‘𝐴) ∈ On
19		rankon 9792	. . . . . . 7 ⊢ (rank‘𝐵) ∈ On
20	18, 19	onun2i 6485	. . . . . 6 ⊢ ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ On
21		eleq2 2820	. . . . . . . . 9 ⊢ (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → ((rank‘𝑥) ∈ 𝑦 ↔ (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
22	21	ralbidv 3175	. . . . . . . 8 ⊢ (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
23		eleq2 2820	. . . . . . . 8 ⊢ (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
24	22, 23	imbi12d 343	. . . . . . 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 3607	. . . . . 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 3987	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘(𝐴 ∪ 𝐵)) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵)))
30		ssun1 4171	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
31		rankssb 9845	. . . . 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 4172	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
34		rankssb 9845	. . . . 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 4185	. . 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 3998	1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104  ∀wral 3059  {crab 3430   ∪ cun 3945   ⊆ wss 3947  ∪ cuni 4907  ∩ cint 4949   “ cima 5678  Oncon0 6363  ‘cfv 6542  𝑅₁cr1 9759  rankcrnk 9760

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-r1 9761  df-rank 9762

This theorem is referenced by:  rankprb  9848  rankopb  9849  rankun  9853  rankaltopb  35255