Theorem hfun 36529

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 36517	. . 3 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
2		elhf2g 36527	. . . . 5 ⊢ (𝐴 ∈ Hf → (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω))
3	2	ibi 269	. . . 4 ⊢ (𝐴 ∈ Hf → (rank‘𝐴) ∈ ω)
4		elhf2g 36527	. . . . 5 ⊢ (𝐵 ∈ Hf → (𝐵 ∈ Hf ↔ (rank‘𝐵) ∈ ω))
5	4	ibi 269	. . . 4 ⊢ (𝐵 ∈ Hf → (rank‘𝐵) ∈ ω)
6		eleq1a 2858	. . . . . 6 ⊢ ((rank‘𝐵) ∈ ω → (((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
7	6	adantl 485	. . . . 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 2768	. . . . . . . . 9 ⊢ (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) ↔ ((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐴))
10	9	biimpi 218	. . . . . . . 8 ⊢ (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → ((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐴))
11	10	eleq1d 2848	. . . . . . 7 ⊢ (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → (((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω ↔ (rank‘𝐴) ∈ ω))
12	11	biimprcd 252	. . . . . 6 ⊢ ((rank‘𝐴) ∈ ω → (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
13	12	adantr 484	. . . . 5 ⊢ (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
14		nnord 7855	. . . . . . 7 ⊢ ((rank‘𝐴) ∈ ω → Ord (rank‘𝐴))
15		nnord 7855	. . . . . . 7 ⊢ ((rank‘𝐵) ∈ ω → Ord (rank‘𝐵))
16		ordtri2or2 6448	. . . . . . 7 ⊢ ((Ord (rank‘𝐴) ∧ Ord (rank‘𝐵)) → ((rank‘𝐴) ⊆ (rank‘𝐵) ∨ (rank‘𝐵) ⊆ (rank‘𝐴)))
17	14, 15, 16	syl2an 605	. . . . . 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 925	. . . . . 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 220	. . . . 5 ⊢ (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → (((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵) ∨ ((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴)))
22	7, 13, 21	mpjaod 871	. . . 4 ⊢ (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω)
23	3, 5, 22	syl2an 605	. . 3 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω)
24	1, 23	eqeltrd 2863	. 2 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (rank‘(𝐴 ∪ 𝐵)) ∈ ω)
25		unexg 7727	. . 3 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 ∪ 𝐵) ∈ V)
26		elhf2g 36527	. . 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 259	1 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 ∪ 𝐵) ∈ Hf )

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1561   ∈ wcel 2143  Vcvv 3455   ∪ cun 3903   ⊆ wss 3905  Ord word 6346  ‘cfv 6522  ωcom 7847  rankcrnk 9722   Hf chf 36523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-reg 9541  ax-inf2 9597

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-ov 7400  df-om 7848  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-er 8679  df-en 8929  df-dom 8930  df-sdom 8931  df-r1 9723  df-rank 9724  df-hf 36524

This theorem is referenced by:  hfadj  36531