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Theorem hfun 35088
Description: The union of two HF sets is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
Assertion
Ref Expression
hfun ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴𝐵) ∈ Hf )

Proof of Theorem hfun
StepHypRef Expression
1 rankung 35076 . . 3 ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
2 elhf2g 35086 . . . . 5 (𝐴 ∈ Hf → (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω))
32ibi 267 . . . 4 (𝐴 ∈ Hf → (rank‘𝐴) ∈ ω)
4 elhf2g 35086 . . . . 5 (𝐵 ∈ Hf → (𝐵 ∈ Hf ↔ (rank‘𝐵) ∈ ω))
54ibi 267 . . . 4 (𝐵 ∈ Hf → (rank‘𝐵) ∈ ω)
6 eleq1a 2829 . . . . . 6 ((rank‘𝐵) ∈ ω → (((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
76adantl 483 . . . . 5 (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → (((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
8 uncom 4152 . . . . . . . . . 10 ((rank‘𝐵) ∪ (rank‘𝐴)) = ((rank‘𝐴) ∪ (rank‘𝐵))
98eqeq1i 2738 . . . . . . . . 9 (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) ↔ ((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐴))
109biimpi 215 . . . . . . . 8 (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → ((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐴))
1110eleq1d 2819 . . . . . . 7 (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → (((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω ↔ (rank‘𝐴) ∈ ω))
1211biimprcd 249 . . . . . 6 ((rank‘𝐴) ∈ ω → (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
1312adantr 482 . . . . 5 (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
14 nnord 7858 . . . . . . 7 ((rank‘𝐴) ∈ ω → Ord (rank‘𝐴))
15 nnord 7858 . . . . . . 7 ((rank‘𝐵) ∈ ω → Ord (rank‘𝐵))
16 ordtri2or2 6460 . . . . . . 7 ((Ord (rank‘𝐴) ∧ Ord (rank‘𝐵)) → ((rank‘𝐴) ⊆ (rank‘𝐵) ∨ (rank‘𝐵) ⊆ (rank‘𝐴)))
1714, 15, 16syl2an 597 . . . . . 6 (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → ((rank‘𝐴) ⊆ (rank‘𝐵) ∨ (rank‘𝐵) ⊆ (rank‘𝐴)))
18 ssequn1 4179 . . . . . . 7 ((rank‘𝐴) ⊆ (rank‘𝐵) ↔ ((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵))
19 ssequn1 4179 . . . . . . 7 ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴))
2018, 19orbi12i 914 . . . . . 6 (((rank‘𝐴) ⊆ (rank‘𝐵) ∨ (rank‘𝐵) ⊆ (rank‘𝐴)) ↔ (((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵) ∨ ((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴)))
2117, 20sylib 217 . . . . 5 (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → (((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵) ∨ ((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴)))
227, 13, 21mpjaod 859 . . . 4 (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω)
233, 5, 22syl2an 597 . . 3 ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω)
241, 23eqeltrd 2834 . 2 ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (rank‘(𝐴𝐵)) ∈ ω)
25 unexg 7731 . . 3 ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴𝐵) ∈ V)
26 elhf2g 35086 . . 3 ((𝐴𝐵) ∈ V → ((𝐴𝐵) ∈ Hf ↔ (rank‘(𝐴𝐵)) ∈ ω))
2725, 26syl 17 . 2 ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → ((𝐴𝐵) ∈ Hf ↔ (rank‘(𝐴𝐵)) ∈ ω))
2824, 27mpbird 257 1 ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴𝐵) ∈ Hf )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  Vcvv 3475  cun 3945  wss 3947  Ord word 6360  cfv 6540  ωcom 7850  rankcrnk 9754   Hf chf 35082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-reg 9583  ax-inf2 9632
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-r1 9755  df-rank 9756  df-hf 35083
This theorem is referenced by:  hfadj  35090
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