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Theorem unwf 8711
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)) ↔ (𝐴𝐵) ∈ (𝑅1 “ On))

Proof of Theorem unwf
StepHypRef Expression
1 r1rankidb 8705 . . . . . . . 8 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
21adantr 480 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
3 ssun1 3809 . . . . . . . 8 (rank‘𝐴) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵))
4 rankdmr1 8702 . . . . . . . . 9 (rank‘𝐴) ∈ dom 𝑅1
5 r1funlim 8667 . . . . . . . . . . . 12 (Fun 𝑅1 ∧ Lim dom 𝑅1)
65simpri 477 . . . . . . . . . . 11 Lim dom 𝑅1
7 limord 5822 . . . . . . . . . . 11 (Lim dom 𝑅1 → Ord dom 𝑅1)
86, 7ax-mp 5 . . . . . . . . . 10 Ord dom 𝑅1
9 rankdmr1 8702 . . . . . . . . . 10 (rank‘𝐵) ∈ dom 𝑅1
10 ordunel 7069 . . . . . . . . . 10 ((Ord dom 𝑅1 ∧ (rank‘𝐴) ∈ dom 𝑅1 ∧ (rank‘𝐵) ∈ dom 𝑅1) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1)
118, 4, 9, 10mp3an 1464 . . . . . . . . 9 ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1
12 r1ord3g 8680 . . . . . . . . 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)))))
134, 11, 12mp2an 708 . . . . . . . 8 ((rank‘𝐴) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵)) → (𝑅1‘(rank‘𝐴)) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
143, 13ax-mp 5 . . . . . . 7 (𝑅1‘(rank‘𝐴)) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵)))
152, 14syl6ss 3648 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → 𝐴 ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
16 r1rankidb 8705 . . . . . . . 8 (𝐵 (𝑅1 “ On) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
1716adantl 481 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
18 ssun2 3810 . . . . . . . 8 (rank‘𝐵) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵))
19 r1ord3g 8680 . . . . . . . . 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)))))
209, 11, 19mp2an 708 . . . . . . . 8 ((rank‘𝐵) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵)) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
2118, 20ax-mp 5 . . . . . . 7 (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵)))
2217, 21syl6ss 3648 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → 𝐵 ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
2315, 22unssd 3822 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝐴𝐵) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
24 fvex 6239 . . . . . 6 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))) ∈ V
2524elpw2 4858 . . . . 5 ((𝐴𝐵) ∈ 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))) ↔ (𝐴𝐵) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
2623, 25sylibr 224 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝐴𝐵) ∈ 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
27 r1sucg 8670 . . . . 5 (((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1 → (𝑅1‘suc ((rank‘𝐴) ∪ (rank‘𝐵))) = 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
2811, 27ax-mp 5 . . . 4 (𝑅1‘suc ((rank‘𝐴) ∪ (rank‘𝐵))) = 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵)))
2926, 28syl6eleqr 2741 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝐴𝐵) ∈ (𝑅1‘suc ((rank‘𝐴) ∪ (rank‘𝐵))))
30 r1elwf 8697 . . 3 ((𝐴𝐵) ∈ (𝑅1‘suc ((rank‘𝐴) ∪ (rank‘𝐵))) → (𝐴𝐵) ∈ (𝑅1 “ On))
3129, 30syl 17 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝐴𝐵) ∈ (𝑅1 “ On))
32 ssun1 3809 . . . 4 𝐴 ⊆ (𝐴𝐵)
33 sswf 8709 . . . 4 (((𝐴𝐵) ∈ (𝑅1 “ On) ∧ 𝐴 ⊆ (𝐴𝐵)) → 𝐴 (𝑅1 “ On))
3432, 33mpan2 707 . . 3 ((𝐴𝐵) ∈ (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
35 ssun2 3810 . . . 4 𝐵 ⊆ (𝐴𝐵)
36 sswf 8709 . . . 4 (((𝐴𝐵) ∈ (𝑅1 “ On) ∧ 𝐵 ⊆ (𝐴𝐵)) → 𝐵 (𝑅1 “ On))
3735, 36mpan2 707 . . 3 ((𝐴𝐵) ∈ (𝑅1 “ On) → 𝐵 (𝑅1 “ On))
3834, 37jca 553 . 2 ((𝐴𝐵) ∈ (𝑅1 “ On) → (𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)))
3931, 38impbii 199 1 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) ↔ (𝐴𝐵) ∈ (𝑅1 “ On))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  cun 3605  wss 3607  𝒫 cpw 4191   cuni 4468  dom cdm 5143  cima 5146  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763  Fun wfun 5920  cfv 5926  𝑅1cr1 8663  rankcrnk 8664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-r1 8665  df-rank 8666
This theorem is referenced by:  prwf  8712  rankunb  8751
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