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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)) ↔ (𝐴𝐵) ∈ (𝑅1 “ On))

Proof of Theorem unwf
StepHypRef Expression
1 r1rankidb 8831 . . . . . . . 8 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
21adantr 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)
65simpri 473 . . . . . . . . . . 11 Lim dom 𝑅1
7 limord 5927 . . . . . . . . . . 11 (Lim dom 𝑅1 → Ord dom 𝑅1)
86, 7ax-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)
118, 4, 9, 10mp3an 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)))))
134, 11, 12mp2an 672 . . . . . . . 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 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)))
1716adantl 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)))))
209, 11, 19mp2an 672 . . . . . . . 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 3764 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → 𝐵 ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
2315, 22unssd 3940 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝐴𝐵) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
24 fvex 6342 . . . . . 6 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))) ∈ V
2524elpw2 4959 . . . . 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 8796 . . . . 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 2861 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝐴𝐵) ∈ (𝑅1‘suc ((rank‘𝐴) ∪ (rank‘𝐵))))
30 r1elwf 8823 . . 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 3927 . . . 4 𝐴 ⊆ (𝐴𝐵)
33 sswf 8835 . . . 4 (((𝐴𝐵) ∈ (𝑅1 “ On) ∧ 𝐴 ⊆ (𝐴𝐵)) → 𝐴 (𝑅1 “ On))
3432, 33mpan2 671 . . 3 ((𝐴𝐵) ∈ (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
35 ssun2 3928 . . . 4 𝐵 ⊆ (𝐴𝐵)
36 sswf 8835 . . . 4 (((𝐴𝐵) ∈ (𝑅1 “ On) ∧ 𝐵 ⊆ (𝐴𝐵)) → 𝐵 (𝑅1 “ On))
3735, 36mpan2 671 . . 3 ((𝐴𝐵) ∈ (𝑅1 “ On) → 𝐵 (𝑅1 “ On))
3834, 37jca 501 . 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 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
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