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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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