Theorem unwf 9695

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 9689	. . . . . . . 8 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2	1	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
3		ssun1 4126	. . . . . . . 8 ⊢ (rank‘𝐴) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵))
4		rankdmr1 9686	. . . . . . . . 9 ⊢ (rank‘𝐴) ∈ dom 𝑅1
5		r1funlim 9651	. . . . . . . . . . . 12 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
6	5	simpri 485	. . . . . . . . . . 11 ⊢ Lim dom 𝑅1
7		limord 6363	. . . . . . . . . . 11 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
8	6, 7	ax-mp 5	. . . . . . . . . 10 ⊢ Ord dom 𝑅1
9		rankdmr1 9686	. . . . . . . . . 10 ⊢ (rank‘𝐵) ∈ dom 𝑅1
10		ordunel 7752	. . . . . . . . . 10 ⊢ ((Ord dom 𝑅1 ∧ (rank‘𝐴) ∈ dom 𝑅1 ∧ (rank‘𝐵) ∈ dom 𝑅1) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1)
11	8, 4, 9, 10	mp3an 1463	. . . . . . . . 9 ⊢ ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1
12		r1ord3g 9664	. . . . . . . . 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 692	. . . . . . . 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 3945	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐴 ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
16		r1rankidb 9689	. . . . . . . 8 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
17	16	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
18		ssun2 4127	. . . . . . . 8 ⊢ (rank‘𝐵) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵))
19		r1ord3g 9664	. . . . . . . . 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 692	. . . . . . . 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 3945	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐵 ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
23	15, 22	unssd 4140	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 ∪ 𝐵) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
24		fvex 6830	. . . . . 6 ⊢ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))) ∈ V
25	24	elpw2 5270	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) ∈ 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))) ↔ (𝐴 ∪ 𝐵) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
26	23, 25	sylibr 234	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 ∪ 𝐵) ∈ 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
27		r1sucg 9654	. . . . 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 2840	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 ∪ 𝐵) ∈ (𝑅1‘suc ((rank‘𝐴) ∪ (rank‘𝐵))))
30		r1elwf 9681	. . 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 4126	. . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
33		sswf 9693	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → 𝐴 ∈ ∪ (𝑅1 “ On))
34	32, 33	mpan2 691	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On))
35		ssun2 4127	. . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
36		sswf 9693	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → 𝐵 ∈ ∪ (𝑅1 “ On))
37	35, 36	mpan2 691	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → 𝐵 ∈ ∪ (𝑅1 “ On))
38	34, 37	jca 511	. 2 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)))
39	31, 38	impbii 209	1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) ↔ (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110   ∪ cun 3898   ⊆ wss 3900  𝒫 cpw 4548  ∪ cuni 4857  dom cdm 5614   “ cima 5617  Ord word 6301  Oncon0 6302  Lim wlim 6303  suc csuc 6304  Fun wfun 6471  ‘cfv 6477  𝑅₁cr1 9647  rankcrnk 9648

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 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-ov 7344  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-r1 9649  df-rank 9650

This theorem is referenced by:  prwf  9696  rankunb  9735  xpwf  44976