Theorem unwf 9241

Description: A binary union is well-founded iff its elements are. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
unwf	⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) ↔ (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))

Proof of Theorem unwf

Step	Hyp	Ref	Expression
1		r1rankidb 9235	. . . . . . . 8 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2	1	adantr 483	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
3		ssun1 4150	. . . . . . . 8 ⊢ (rank‘𝐴) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵))
4		rankdmr1 9232	. . . . . . . . 9 ⊢ (rank‘𝐴) ∈ dom 𝑅1
5		r1funlim 9197	. . . . . . . . . . . 12 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
6	5	simpri 488	. . . . . . . . . . 11 ⊢ Lim dom 𝑅1
7		limord 6252	. . . . . . . . . . 11 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
8	6, 7	ax-mp 5	. . . . . . . . . 10 ⊢ Ord dom 𝑅1
9		rankdmr1 9232	. . . . . . . . . 10 ⊢ (rank‘𝐵) ∈ dom 𝑅1
10		ordunel 7544	. . . . . . . . . 10 ⊢ ((Ord dom 𝑅1 ∧ (rank‘𝐴) ∈ dom 𝑅1 ∧ (rank‘𝐵) ∈ dom 𝑅1) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1)
11	8, 4, 9, 10	mp3an 1457	. . . . . . . . 9 ⊢ ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1
12		r1ord3g 9210	. . . . . . . . 9 ⊢ (((rank‘𝐴) ∈ dom 𝑅1 ∧ ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1) → ((rank‘𝐴) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵)) → (𝑅1‘(rank‘𝐴)) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵)))))
13	4, 11, 12	mp2an 690	. . . . . . . 8 ⊢ ((rank‘𝐴) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵)) → (𝑅1‘(rank‘𝐴)) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
14	3, 13	ax-mp 5	. . . . . . 7 ⊢ (𝑅1‘(rank‘𝐴)) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵)))
15	2, 14	sstrdi 3981	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐴 ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
16		r1rankidb 9235	. . . . . . . 8 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
17	16	adantl 484	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
18		ssun2 4151	. . . . . . . 8 ⊢ (rank‘𝐵) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵))
19		r1ord3g 9210	. . . . . . . . 9 ⊢ (((rank‘𝐵) ∈ dom 𝑅1 ∧ ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1) → ((rank‘𝐵) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵)) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵)))))
20	9, 11, 19	mp2an 690	. . . . . . . 8 ⊢ ((rank‘𝐵) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵)) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
21	18, 20	ax-mp 5	. . . . . . 7 ⊢ (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵)))
22	17, 21	sstrdi 3981	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐵 ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
23	15, 22	unssd 4164	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 ∪ 𝐵) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
24		fvex 6685	. . . . . 6 ⊢ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))) ∈ V
25	24	elpw2 5250	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) ∈ 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))) ↔ (𝐴 ∪ 𝐵) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
26	23, 25	sylibr 236	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 ∪ 𝐵) ∈ 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
27		r1sucg 9200	. . . . 5 ⊢ (((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1 → (𝑅1‘suc ((rank‘𝐴) ∪ (rank‘𝐵))) = 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
28	11, 27	ax-mp 5	. . . 4 ⊢ (𝑅1‘suc ((rank‘𝐴) ∪ (rank‘𝐵))) = 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵)))
29	26, 28	eleqtrrdi 2926	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 ∪ 𝐵) ∈ (𝑅1‘suc ((rank‘𝐴) ∪ (rank‘𝐵))))
30		r1elwf 9227	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ (𝑅1‘suc ((rank‘𝐴) ∪ (rank‘𝐵))) → (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
31	29, 30	syl 17	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
32		ssun1 4150	. . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
33		sswf 9239	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → 𝐴 ∈ ∪ (𝑅1 “ On))
34	32, 33	mpan2 689	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On))
35		ssun2 4151	. . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
36		sswf 9239	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → 𝐵 ∈ ∪ (𝑅1 “ On))
37	35, 36	mpan2 689	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → 𝐵 ∈ ∪ (𝑅1 “ On))
38	34, 37	jca 514	. 2 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)))
39	31, 38	impbii 211	1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) ↔ (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ∪ cun 3936   ⊆ wss 3938  𝒫 cpw 4541  ∪ cuni 4840  dom cdm 5557   “ cima 5560  Ord word 6192  Oncon0 6193  Lim wlim 6194  suc csuc 6195  Fun wfun 6351  ‘cfv 6357  𝑅₁cr1 9193  rankcrnk 9194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-r1 9195  df-rank 9196

This theorem is referenced by:  prwf  9242  rankunb  9281