Theorem unwf 9825

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 9819	. . . . . . . 8 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2	1	adantr 486	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
3		ssun1 4175	. . . . . . . 8 ⊢ (rank‘𝐴) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵))
4		rankdmr1 9816	. . . . . . . . 9 ⊢ (rank‘𝐴) ∈ dom 𝑅1
5		r1dmlim 9779	. . . . . . . . . . 11 ⊢ Lim dom 𝑅1
6		limord 6427	. . . . . . . . . . 11 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
7	5, 6	ax-mp 5	. . . . . . . . . 10 ⊢ Ord dom 𝑅1
8		rankdmr1 9816	. . . . . . . . . 10 ⊢ (rank‘𝐵) ∈ dom 𝑅1
9		ordunel 7842	. . . . . . . . . 10 ⊢ ((Ord dom 𝑅1 ∧ (rank‘𝐴) ∈ dom 𝑅1 ∧ (rank‘𝐵) ∈ dom 𝑅1) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1)
10	7, 4, 8, 9	mp3an 1490	. . . . . . . . 9 ⊢ ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1
11		r1ord3g 9793	. . . . . . . . 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)))))
12	4, 10, 11	mp2an 705	. . . . . . . 8 ⊢ ((rank‘𝐴) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵)) → (𝑅1‘(rank‘𝐴)) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
13	3, 12	ax-mp 5	. . . . . . 7 ⊢ (𝑅1‘(rank‘𝐴)) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵)))
14	2, 13	sstrdi 3921	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐴 ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
15		r1rankidb 9819	. . . . . . . 8 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
16	15	adantl 487	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
17		ssun2 4176	. . . . . . . 8 ⊢ (rank‘𝐵) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵))
18		r1ord3g 9793	. . . . . . . . 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)))))
19	8, 10, 18	mp2an 705	. . . . . . . 8 ⊢ ((rank‘𝐵) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵)) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
20	17, 19	ax-mp 5	. . . . . . 7 ⊢ (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵)))
21	16, 20	sstrdi 3921	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐵 ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
22	14, 21	unssd 4189	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 ∪ 𝐵) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
23		fvex 6903	. . . . . 6 ⊢ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))) ∈ V
24	23	elpw2 5296	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) ∈ 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))) ↔ (𝐴 ∪ 𝐵) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
25	22, 24	sylibr 237	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 ∪ 𝐵) ∈ 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
26		r1sucg 9783	. . . . 5 ⊢ (((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1 → (𝑅1‘suc ((rank‘𝐴) ∪ (rank‘𝐵))) = 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
27	10, 26	ax-mp 5	. . . 4 ⊢ (𝑅1‘suc ((rank‘𝐴) ∪ (rank‘𝐵))) = 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵)))
28	25, 27	eleqtrrdi 2871	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 ∪ 𝐵) ∈ (𝑅1‘suc ((rank‘𝐴) ∪ (rank‘𝐵))))
29		r1elwf 9811	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ (𝑅1‘suc ((rank‘𝐴) ∪ (rank‘𝐵))) → (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
30	28, 29	syl 18	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
31		ssun1 4175	. . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
32		sswf 9823	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → 𝐴 ∈ ∪ (𝑅1 “ On))
33	31, 32	mpan2 704	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On))
34		ssun2 4176	. . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
35		sswf 9823	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → 𝐵 ∈ ∪ (𝑅1 “ On))
36	34, 35	mpan2 704	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → 𝐵 ∈ ∪ (𝑅1 “ On))
37	33, 36	jca 521	. 2 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)))
38	30, 37	impbii 212	1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) ↔ (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3896   ∪ cun 4148  𝒫 cpw 4556  ∪ cuni 4866  dom cdm 5651   “ cima 5654  Ord word 6364  Oncon0 6365  Lim wlim 6366  suc csuc 6367  ‘cfv 6543  𝑅₁cr1 9773  rankcrnk 9774

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-om 7882  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-r1 9775  df-rank 9776

This theorem is used by:  prwf  9826  rankunb  9871  xpwf  46090