Theorem hfun 36139

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 36127	. . 3 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
2		elhf2g 36137	. . . . 5 ⊢ (𝐴 ∈ Hf → (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω))
3	2	ibi 267	. . . 4 ⊢ (𝐴 ∈ Hf → (rank‘𝐴) ∈ ω)
4		elhf2g 36137	. . . . 5 ⊢ (𝐵 ∈ Hf → (𝐵 ∈ Hf ↔ (rank‘𝐵) ∈ ω))
5	4	ibi 267	. . . 4 ⊢ (𝐵 ∈ Hf → (rank‘𝐵) ∈ ω)
6		eleq1a 2823	. . . . . 6 ⊢ ((rank‘𝐵) ∈ ω → (((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
7	6	adantl 481	. . . . 5 ⊢ (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → (((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
8		uncom 4117	. . . . . . . . . 10 ⊢ ((rank‘𝐵) ∪ (rank‘𝐴)) = ((rank‘𝐴) ∪ (rank‘𝐵))
9	8	eqeq1i 2734	. . . . . . . . 9 ⊢ (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) ↔ ((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐴))
10	9	biimpi 216	. . . . . . . 8 ⊢ (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → ((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐴))
11	10	eleq1d 2813	. . . . . . 7 ⊢ (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → (((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω ↔ (rank‘𝐴) ∈ ω))
12	11	biimprcd 250	. . . . . 6 ⊢ ((rank‘𝐴) ∈ ω → (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
13	12	adantr 480	. . . . 5 ⊢ (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
14		nnord 7830	. . . . . . 7 ⊢ ((rank‘𝐴) ∈ ω → Ord (rank‘𝐴))
15		nnord 7830	. . . . . . 7 ⊢ ((rank‘𝐵) ∈ ω → Ord (rank‘𝐵))
16		ordtri2or2 6421	. . . . . . 7 ⊢ ((Ord (rank‘𝐴) ∧ Ord (rank‘𝐵)) → ((rank‘𝐴) ⊆ (rank‘𝐵) ∨ (rank‘𝐵) ⊆ (rank‘𝐴)))
17	14, 15, 16	syl2an 596	. . . . . 6 ⊢ (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → ((rank‘𝐴) ⊆ (rank‘𝐵) ∨ (rank‘𝐵) ⊆ (rank‘𝐴)))
18		ssequn1 4145	. . . . . . 7 ⊢ ((rank‘𝐴) ⊆ (rank‘𝐵) ↔ ((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵))
19		ssequn1 4145	. . . . . . 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 218	. . . . 5 ⊢ (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → (((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵) ∨ ((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴)))
22	7, 13, 21	mpjaod 860	. . . 4 ⊢ (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω)
23	3, 5, 22	syl2an 596	. . 3 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω)
24	1, 23	eqeltrd 2828	. 2 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (rank‘(𝐴 ∪ 𝐵)) ∈ ω)
25		unexg 7699	. . 3 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 ∪ 𝐵) ∈ V)
26		elhf2g 36137	. . 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 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ∪ cun 3909   ⊆ wss 3911  Ord word 6319  ‘cfv 6499  ωcom 7822  rankcrnk 9692   Hf chf 36133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-reg 9521  ax-inf2 9570

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-r1 9693  df-rank 9694  df-hf 36134

This theorem is referenced by:  hfadj  36141