Theorem hfunOLD 9919

Description: Obsolete version of hfun 9918 as of 17-Sep-2026. (Contributed by Scott Fenton, 15-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
hfunOLD	⊢ ((𝐴 ∈ HF ∧ 𝐵 ∈ HF ) → (𝐴 ∪ 𝐵) ∈ HF )

Proof of Theorem hfunOLD

Step	Hyp	Ref	Expression
1		rankung 9873	. . 3 ⊢ ((𝐴 ∈ HF ∧ 𝐵 ∈ HF ) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
2		elhf2g 9911	. . . . 5 ⊢ (𝐴 ∈ HF → (𝐴 ∈ HF ↔ (rank‘𝐴) ∈ ω))
3	2	ibi 270	. . . 4 ⊢ (𝐴 ∈ HF → (rank‘𝐴) ∈ ω)
4		elhf2g 9911	. . . . 5 ⊢ (𝐵 ∈ HF → (𝐵 ∈ HF ↔ (rank‘𝐵) ∈ ω))
5	4	ibi 270	. . . 4 ⊢ (𝐵 ∈ HF → (rank‘𝐵) ∈ ω)
6		eleq1a 2855	. . . . . 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 4157	. . . . . . . . . 10 ⊢ ((rank‘𝐵) ∪ (rank‘𝐴)) = ((rank‘𝐴) ∪ (rank‘𝐵))
9	8	eqeq1i 2765	. . . . . . . . 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 2845	. . . . . . 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 7885	. . . . . . 7 ⊢ ((rank‘𝐴) ∈ ω → Ord (rank‘𝐴))
15		nnord 7885	. . . . . . 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 608	. . . . . 6 ⊢ (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → ((rank‘𝐴) ⊆ (rank‘𝐵) ∨ (rank‘𝐵) ⊆ (rank‘𝐴)))
18		ssequn1 4184	. . . . . . 7 ⊢ ((rank‘𝐴) ⊆ (rank‘𝐵) ↔ ((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵))
19		ssequn1 4184	. . . . . . 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 2860	. 2 ⊢ ((𝐴 ∈ HF ∧ 𝐵 ∈ HF ) → (rank‘(𝐴 ∪ 𝐵)) ∈ ω)
25		unexg 7760	. . 3 ⊢ ((𝐴 ∈ HF ∧ 𝐵 ∈ HF ) → (𝐴 ∪ 𝐵) ∈ V)
26		elhf2g 9911	. . 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 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895   ∪ cun 4149  Ord word 6361  ‘cfv 6538  ωcom 7877  rankcrnk 9767   HF chf 9904

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-reg 9586  ax-inf2 9642

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  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 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-r1 9768  df-rank 9769  df-hf 9905

This theorem is used by: (None)