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Theorem unwf 8556
 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 8550 . . . . . . . 8 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
21adantr 480 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
3 ssun1 3738 . . . . . . . 8 (rank‘𝐴) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵))
4 rankdmr1 8547 . . . . . . . . 9 (rank‘𝐴) ∈ dom 𝑅1
5 r1funlim 8512 . . . . . . . . . . . 12 (Fun 𝑅1 ∧ Lim dom 𝑅1)
65simpri 477 . . . . . . . . . . 11 Lim dom 𝑅1
7 limord 5701 . . . . . . . . . . 11 (Lim dom 𝑅1 → Ord dom 𝑅1)
86, 7ax-mp 5 . . . . . . . . . 10 Ord dom 𝑅1
9 rankdmr1 8547 . . . . . . . . . 10 (rank‘𝐵) ∈ dom 𝑅1
10 ordunel 6919 . . . . . . . . . 10 ((Ord dom 𝑅1 ∧ (rank‘𝐴) ∈ dom 𝑅1 ∧ (rank‘𝐵) ∈ dom 𝑅1) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1)
118, 4, 9, 10mp3an 1416 . . . . . . . . 9 ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ dom 𝑅1
12 r1ord3g 8525 . . . . . . . . 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 704 . . . . . . . 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 3580 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → 𝐴 ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
16 r1rankidb 8550 . . . . . . . 8 (𝐵 (𝑅1 “ On) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
1716adantl 481 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
18 ssun2 3739 . . . . . . . 8 (rank‘𝐵) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵))
19 r1ord3g 8525 . . . . . . . . 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 704 . . . . . . . 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 3580 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → 𝐵 ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
2315, 22unssd 3751 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝐴𝐵) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
24 fvex 6113 . . . . . 6 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))) ∈ V
2524elpw2 4755 . . . . 5 ((𝐴𝐵) ∈ 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))) ↔ (𝐴𝐵) ⊆ (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
2623, 25sylibr 223 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝐴𝐵) ∈ 𝒫 (𝑅1‘((rank‘𝐴) ∪ (rank‘𝐵))))
27 r1sucg 8515 . . . . 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 2699 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝐴𝐵) ∈ (𝑅1‘suc ((rank‘𝐴) ∪ (rank‘𝐵))))
30 r1elwf 8542 . . 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 3738 . . . 4 𝐴 ⊆ (𝐴𝐵)
33 sswf 8554 . . . 4 (((𝐴𝐵) ∈ (𝑅1 “ On) ∧ 𝐴 ⊆ (𝐴𝐵)) → 𝐴 (𝑅1 “ On))
3432, 33mpan2 703 . . 3 ((𝐴𝐵) ∈ (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
35 ssun2 3739 . . . 4 𝐵 ⊆ (𝐴𝐵)
36 sswf 8554 . . . 4 (((𝐴𝐵) ∈ (𝑅1 “ On) ∧ 𝐵 ⊆ (𝐴𝐵)) → 𝐵 (𝑅1 “ On))
3735, 36mpan2 703 . . 3 ((𝐴𝐵) ∈ (𝑅1 “ On) → 𝐵 (𝑅1 “ On))
3834, 37jca 553 . 2 ((𝐴𝐵) ∈ (𝑅1 “ On) → (𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)))
3931, 38impbii 198 1 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) ↔ (𝐴𝐵) ∈ (𝑅1 “ On))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  dom cdm 5038   “ cima 5041  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  Fun wfun 5798  ‘cfv 5804  𝑅1cr1 8508  rankcrnk 8509 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510  df-rank 8511 This theorem is referenced by:  prwf  8557  rankunb  8596
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