Theorem rankunb 9845

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 9805	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) ↔ (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
2		rankval3b 9821	. . . . . . 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 2820	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ 𝑥 ∈ ∩ {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦}))
5		vex 3479	. . . . . 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 4149	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
9		rankelb 9819	. . . . . . . . 9 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
10		elun1 4177	. . . . . . . . 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 9819	. . . . . . . . 9 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐵 → (rank‘𝑥) ∈ (rank‘𝐵)))
13		elun2 4178	. . . . . . . . 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 954	. . . . . . 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 3146	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))
18		rankon 9790	. . . . . . 7 ⊢ (rank‘𝐴) ∈ On
19		rankon 9790	. . . . . . 7 ⊢ (rank‘𝐵) ∈ On
20	18, 19	onun2i 6487	. . . . . 6 ⊢ ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ On
21		eleq2 2823	. . . . . . . . 9 ⊢ (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → ((rank‘𝑥) ∈ 𝑦 ↔ (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
22	21	ralbidv 3178	. . . . . . . 8 ⊢ (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
23		eleq2 2823	. . . . . . . 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 3609	. . . . . 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 3989	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘(𝐴 ∪ 𝐵)) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵)))
30		ssun1 4173	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
31		rankssb 9843	. . . . 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 4174	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
34		rankssb 9843	. . . . 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 4187	. . 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 4000	1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  ∀wral 3062  {crab 3433   ∪ cun 3947   ⊆ wss 3949  ∪ cuni 4909  ∩ cint 4951   “ cima 5680  Oncon0 6365  ‘cfv 6544  𝑅₁cr1 9757  rankcrnk 9758

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-r1 9759  df-rank 9760

This theorem is referenced by:  rankprb  9846  rankopb  9847  rankun  9851  rankaltopb  34951