Theorem hfun 36711

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 36699	. . 3 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
2		elhf2g 36709	. . . . 5 ⊢ (𝐴 ∈ Hf → (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω))
3	2	ibi 270	. . . 4 ⊢ (𝐴 ∈ Hf → (rank‘𝐴) ∈ ω)
4		elhf2g 36709	. . . . 5 ⊢ (𝐵 ∈ Hf → (𝐵 ∈ Hf ↔ (rank‘𝐵) ∈ ω))
5	4	ibi 270	. . . 4 ⊢ (𝐵 ∈ Hf → (rank‘𝐵) ∈ ω)
6		eleq1a 2860	. . . . . 6 ⊢ ((rank‘𝐵) ∈ ω → (((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
7	6	adantl 487	. . . . 5 ⊢ (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → (((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
8		uncom 4112	. . . . . . . . . 10 ⊢ ((rank‘𝐵) ∪ (rank‘𝐴)) = ((rank‘𝐴) ∪ (rank‘𝐵))
9	8	eqeq1i 2770	. . . . . . . . 9 ⊢ (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) ↔ ((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐴))
10	9	biimpi 219	. . . . . . . 8 ⊢ (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → ((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐴))
11	10	eleq1d 2850	. . . . . . 7 ⊢ (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → (((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω ↔ (rank‘𝐴) ∈ ω))
12	11	biimprcd 253	. . . . . 6 ⊢ ((rank‘𝐴) ∈ ω → (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
13	12	adantr 486	. . . . 5 ⊢ (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
14		nnord 7876	. . . . . . 7 ⊢ ((rank‘𝐴) ∈ ω → Ord (rank‘𝐴))
15		nnord 7876	. . . . . . 7 ⊢ ((rank‘𝐵) ∈ ω → Ord (rank‘𝐵))
16		ordtri2or2 6466	. . . . . . 7 ⊢ ((Ord (rank‘𝐴) ∧ Ord (rank‘𝐵)) → ((rank‘𝐴) ⊆ (rank‘𝐵) ∨ (rank‘𝐵) ⊆ (rank‘𝐴)))
17	14, 15, 16	syl2an 608	. . . . . 6 ⊢ (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → ((rank‘𝐴) ⊆ (rank‘𝐵) ∨ (rank‘𝐵) ⊆ (rank‘𝐴)))
18		ssequn1 4139	. . . . . . 7 ⊢ ((rank‘𝐴) ⊆ (rank‘𝐵) ↔ ((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵))
19		ssequn1 4139	. . . . . . 7 ⊢ ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴))
20	18, 19	orbi12i 928	. . . . . 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 221	. . . . 5 ⊢ (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → (((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵) ∨ ((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴)))
22	7, 13, 21	mpjaod 874	. . . 4 ⊢ (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω)
23	3, 5, 22	syl2an 608	. . 3 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω)
24	1, 23	eqeltrd 2865	. 2 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (rank‘(𝐴 ∪ 𝐵)) ∈ ω)
25		unexg 7751	. . 3 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 ∪ 𝐵) ∈ V)
26		elhf2g 36709	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → ((𝐴 ∪ 𝐵) ∈ Hf ↔ (rank‘(𝐴 ∪ 𝐵)) ∈ ω))
27	25, 26	syl 18	. 2 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → ((𝐴 ∪ 𝐵) ∈ Hf ↔ (rank‘(𝐴 ∪ 𝐵)) ∈ ω))
28	24, 27	mpbird 260	1 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 ∪ 𝐵) ∈ Hf )

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146  Vcvv 3457   ∪ cun 3904   ⊆ wss 3906  Ord word 6363  ‘cfv 6540  ωcom 7868  rankcrnk 9742   Hf chf 36705

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-reg 9561  ax-inf2 9617

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-r1 9743  df-rank 9744  df-hf 36706

This theorem is used by:  hfadj  36713