Theorem unwf 9090

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 9084	. . . . . . . 8 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2	1	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
3		ssun1 4073	. . . . . . . 8 ⊢ (rank‘𝐴) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵))
4		rankdmr1 9081	. . . . . . . . 9 ⊢ (rank‘𝐴) ∈ dom 𝑅1
5		r1funlim 9046	. . . . . . . . . . . 12 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
6	5	simpri 486	. . . . . . . . . . 11 ⊢ Lim dom 𝑅1
7		limord 6130	. . . . . . . . . . 11 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
8	6, 7	ax-mp 5	. . . . . . . . . 10 ⊢ Ord dom 𝑅1
9		rankdmr1 9081	. . . . . . . . . 10 ⊢ (rank‘𝐵) ∈ dom 𝑅1
10		ordunel 7403	. . . . . . . . . 10 ⊢ ((Ord dom 𝑅1 ∧ (rank‘𝐴) ∈ dom 𝑅1 ∧ (rank‘𝐵) ∈ dom 𝑅1) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1)
11	8, 4, 9, 10	mp3an 1453	. . . . . . . . 9 ⊢ ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1
12		r1ord3g 9059	. . . . . . . . 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 688	. . . . . . . 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	syl6ss 3905	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐴 ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
16		r1rankidb 9084	. . . . . . . 8 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
17	16	adantl 482	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
18		ssun2 4074	. . . . . . . 8 ⊢ (rank‘𝐵) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵))
19		r1ord3g 9059	. . . . . . . . 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 688	. . . . . . . 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	syl6ss 3905	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐵 ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
23	15, 22	unssd 4087	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 ∪ 𝐵) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
24		fvex 6556	. . . . . 6 ⊢ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))) ∈ V
25	24	elpw2 5144	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) ∈ 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))) ↔ (𝐴 ∪ 𝐵) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
26	23, 25	sylibr 235	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 ∪ 𝐵) ∈ 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
27		r1sucg 9049	. . . . 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	syl6eleqr 2894	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 ∪ 𝐵) ∈ (𝑅1‘suc ((rank‘𝐴) ∪ (rank‘𝐵))))
30		r1elwf 9076	. . 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 4073	. . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
33		sswf 9088	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → 𝐴 ∈ ∪ (𝑅1 “ On))
34	32, 33	mpan2 687	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On))
35		ssun2 4074	. . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
36		sswf 9088	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → 𝐵 ∈ ∪ (𝑅1 “ On))
37	35, 36	mpan2 687	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → 𝐵 ∈ ∪ (𝑅1 “ On))
38	34, 37	jca 512	. 2 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)))
39	31, 38	impbii 210	1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) ↔ (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522   ∈ wcel 2081   ∪ cun 3861   ⊆ wss 3863  𝒫 cpw 4457  ∪ cuni 4749  dom cdm 5448   “ cima 5451  Ord word 6070  Oncon0 6071  Lim wlim 6072  suc csuc 6073  Fun wfun 6224  ‘cfv 6230  𝑅₁cr1 9042  rankcrnk 9043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-int 4787  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-om 7442  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-r1 9044  df-rank 9045

This theorem is referenced by:  prwf  9091  rankunb  9130