Theorem unwf 8837

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 8831	. . . . . . . 8 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2	1	adantr 466	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
3		ssun1 3927	. . . . . . . 8 ⊢ (rank‘𝐴) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵))
4		rankdmr1 8828	. . . . . . . . 9 ⊢ (rank‘𝐴) ∈ dom 𝑅1
5		r1funlim 8793	. . . . . . . . . . . 12 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
6	5	simpri 473	. . . . . . . . . . 11 ⊢ Lim dom 𝑅1
7		limord 5927	. . . . . . . . . . 11 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
8	6, 7	ax-mp 5	. . . . . . . . . 10 ⊢ Ord dom 𝑅1
9		rankdmr1 8828	. . . . . . . . . 10 ⊢ (rank‘𝐵) ∈ dom 𝑅1
10		ordunel 7174	. . . . . . . . . 10 ⊢ ((Ord dom 𝑅1 ∧ (rank‘𝐴) ∈ dom 𝑅1 ∧ (rank‘𝐵) ∈ dom 𝑅1) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1)
11	8, 4, 9, 10	mp3an 1572	. . . . . . . . 9 ⊢ ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1
12		r1ord3g 8806	. . . . . . . . 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 672	. . . . . . . 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 3764	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐴 ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
16		r1rankidb 8831	. . . . . . . 8 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
17	16	adantl 467	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
18		ssun2 3928	. . . . . . . 8 ⊢ (rank‘𝐵) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵))
19		r1ord3g 8806	. . . . . . . . 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 672	. . . . . . . 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 3764	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 𝐵 ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
23	15, 22	unssd 3940	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 ∪ 𝐵) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
24		fvex 6342	. . . . . 6 ⊢ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))) ∈ V
25	24	elpw2 4959	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) ∈ 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))) ↔ (𝐴 ∪ 𝐵) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
26	23, 25	sylibr 224	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 ∪ 𝐵) ∈ 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
27		r1sucg 8796	. . . . 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 2861	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 ∪ 𝐵) ∈ (𝑅1‘suc ((rank‘𝐴) ∪ (rank‘𝐵))))
30		r1elwf 8823	. . 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 3927	. . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
33		sswf 8835	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → 𝐴 ∈ ∪ (𝑅1 “ On))
34	32, 33	mpan2 671	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On))
35		ssun2 3928	. . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
36		sswf 8835	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → 𝐵 ∈ ∪ (𝑅1 “ On))
37	35, 36	mpan2 671	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → 𝐵 ∈ ∪ (𝑅1 “ On))
38	34, 37	jca 501	. 2 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)))
39	31, 38	impbii 199	1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) ↔ (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145   ∪ cun 3721   ⊆ wss 3723  𝒫 cpw 4297  ∪ cuni 4574  dom cdm 5249   “ cima 5252  Ord word 5865  Oncon0 5866  Lim wlim 5867  suc csuc 5868  Fun wfun 6025  ‘cfv 6031  𝑅₁cr1 8789  rankcrnk 8790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-r1 8791  df-rank 8792

This theorem is referenced by:  prwf  8838  rankunb  8877