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Theorem unwf 9223
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 9217 . . . . . . . 8 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
21adantr 484 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
3 ssun1 4099 . . . . . . . 8 (rank‘𝐴) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵))
4 rankdmr1 9214 . . . . . . . . 9 (rank‘𝐴) ∈ dom 𝑅1
5 r1funlim 9179 . . . . . . . . . . . 12 (Fun 𝑅1 ∧ Lim dom 𝑅1)
65simpri 489 . . . . . . . . . . 11 Lim dom 𝑅1
7 limord 6218 . . . . . . . . . . 11 (Lim dom 𝑅1 → Ord dom 𝑅1)
86, 7ax-mp 5 . . . . . . . . . 10 Ord dom 𝑅1
9 rankdmr1 9214 . . . . . . . . . 10 (rank‘𝐵) ∈ dom 𝑅1
10 ordunel 7522 . . . . . . . . . 10 ((Ord dom 𝑅1 ∧ (rank‘𝐴) ∈ dom 𝑅1 ∧ (rank‘𝐵) ∈ dom 𝑅1) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1)
118, 4, 9, 10mp3an 1458 . . . . . . . . 9 ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1
12 r1ord3g 9192 . . . . . . . . 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 691 . . . . . . . 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, 14sstrdi 3927 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → 𝐴 ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
16 r1rankidb 9217 . . . . . . . 8 (𝐵 (𝑅1 “ On) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
1716adantl 485 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
18 ssun2 4100 . . . . . . . 8 (rank‘𝐵) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵))
19 r1ord3g 9192 . . . . . . . . 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 691 . . . . . . . 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, 21sstrdi 3927 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → 𝐵 ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
2315, 22unssd 4113 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝐴𝐵) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
24 fvex 6658 . . . . . 6 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))) ∈ V
2524elpw2 5212 . . . . 5 ((𝐴𝐵) ∈ 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))) ↔ (𝐴𝐵) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
2623, 25sylibr 237 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝐴𝐵) ∈ 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
27 r1sucg 9182 . . . . 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, 28eleqtrrdi 2901 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝐴𝐵) ∈ (𝑅1‘suc ((rank‘𝐴) ∪ (rank‘𝐵))))
30 r1elwf 9209 . . 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 4099 . . . 4 𝐴 ⊆ (𝐴𝐵)
33 sswf 9221 . . . 4 (((𝐴𝐵) ∈ (𝑅1 “ On) ∧ 𝐴 ⊆ (𝐴𝐵)) → 𝐴 (𝑅1 “ On))
3432, 33mpan2 690 . . 3 ((𝐴𝐵) ∈ (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
35 ssun2 4100 . . . 4 𝐵 ⊆ (𝐴𝐵)
36 sswf 9221 . . . 4 (((𝐴𝐵) ∈ (𝑅1 “ On) ∧ 𝐵 ⊆ (𝐴𝐵)) → 𝐵 (𝑅1 “ On))
3735, 36mpan2 690 . . 3 ((𝐴𝐵) ∈ (𝑅1 “ On) → 𝐵 (𝑅1 “ On))
3834, 37jca 515 . 2 ((𝐴𝐵) ∈ (𝑅1 “ On) → (𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)))
3931, 38impbii 212 1 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) ↔ (𝐴𝐵) ∈ (𝑅1 “ On))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  cun 3879  wss 3881  𝒫 cpw 4497   cuni 4800  dom cdm 5519  cima 5522  Ord word 6158  Oncon0 6159  Lim wlim 6160  suc csuc 6161  Fun wfun 6318  cfv 6324  𝑅1cr1 9175  rankcrnk 9176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-r1 9177  df-rank 9178
This theorem is referenced by:  prwf  9224  rankunb  9263
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