Theorem hfun 35150

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 ) → (𝐴 ∪ 𝐵) ∈ Hf )

Proof of Theorem hfun

Step	Hyp	Ref	Expression
1		rankung 35138	. . 3 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
2		elhf2g 35148	. . . . 5 ⊢ (𝐴 ∈ Hf → (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω))
3	2	ibi 267	. . . 4 ⊢ (𝐴 ∈ Hf → (rank‘𝐴) ∈ ω)
4		elhf2g 35148	. . . . 5 ⊢ (𝐵 ∈ Hf → (𝐵 ∈ Hf ↔ (rank‘𝐵) ∈ ω))
5	4	ibi 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))
7	6	adantl 483	. . . . 5 ⊢ (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → (((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
8		uncom 4154	. . . . . . . . . 10 ⊢ ((rank‘𝐵) ∪ (rank‘𝐴)) = ((rank‘𝐴) ∪ (rank‘𝐵))
9	8	eqeq1i 2738	. . . . . . . . 9 ⊢ (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) ↔ ((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐴))
10	9	biimpi 215	. . . . . . . 8 ⊢ (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → ((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐴))
11	10	eleq1d 2819	. . . . . . 7 ⊢ (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → (((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω ↔ (rank‘𝐴) ∈ ω))
12	11	biimprcd 249	. . . . . 6 ⊢ ((rank‘𝐴) ∈ ω → (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
13	12	adantr 482	. . . . 5 ⊢ (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
14		nnord 7863	. . . . . . 7 ⊢ ((rank‘𝐴) ∈ ω → Ord (rank‘𝐴))
15		nnord 7863	. . . . . . 7 ⊢ ((rank‘𝐵) ∈ ω → Ord (rank‘𝐵))
16		ordtri2or2 6464	. . . . . . 7 ⊢ ((Ord (rank‘𝐴) ∧ Ord (rank‘𝐵)) → ((rank‘𝐴) ⊆ (rank‘𝐵) ∨ (rank‘𝐵) ⊆ (rank‘𝐴)))
17	14, 15, 16	syl2an 597	. . . . . 6 ⊢ (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → ((rank‘𝐴) ⊆ (rank‘𝐵) ∨ (rank‘𝐵) ⊆ (rank‘𝐴)))
18		ssequn1 4181	. . . . . . 7 ⊢ ((rank‘𝐴) ⊆ (rank‘𝐵) ↔ ((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵))
19		ssequn1 4181	. . . . . . 7 ⊢ ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴))
20	18, 19	orbi12i 914	. . . . . 6 ⊢ (((rank‘𝐴) ⊆ (rank‘𝐵) ∨ (rank‘𝐵) ⊆ (rank‘𝐴)) ↔ (((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵) ∨ ((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴)))
21	17, 20	sylib 217	. . . . 5 ⊢ (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → (((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵) ∨ ((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴)))
22	7, 13, 21	mpjaod 859	. . . 4 ⊢ (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω)
23	3, 5, 22	syl2an 597	. . 3 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω)
24	1, 23	eqeltrd 2834	. 2 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (rank‘(𝐴 ∪ 𝐵)) ∈ ω)
25		unexg 7736	. . 3 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 ∪ 𝐵) ∈ V)
26		elhf2g 35148	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → ((𝐴 ∪ 𝐵) ∈ Hf ↔ (rank‘(𝐴 ∪ 𝐵)) ∈ ω))
27	25, 26	syl 17	. 2 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → ((𝐴 ∪ 𝐵) ∈ Hf ↔ (rank‘(𝐴 ∪ 𝐵)) ∈ ω))
28	24, 27	mpbird 257	1 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 ∪ 𝐵) ∈ Hf )

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∪ cun 3947   ⊆ wss 3949  Ord word 6364  ‘cfv 6544  ωcom 7855  rankcrnk 9758   Hf chf 35144

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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-reg 9587  ax-inf2 9636

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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-r1 9759  df-rank 9760  df-hf 35145

This theorem is referenced by:  hfadj  35152