Theorem bdayimaon 28050

Description: Lemma for full-eta properties. The successor of the union of the image of the birthday function under a set is an ordinal. (Contributed by Scott Fenton, 20-Aug-2011.)

Assertion

Ref	Expression
bdayimaon	⊢ (𝐴 ∈ 𝑉 → suc ∪ (bday “ 𝐴) ∈ On)

Proof of Theorem bdayimaon

Step	Hyp	Ref	Expression
1		bdayfo 28034	. . . . . 6 ⊢ bday:No–onto→On
2		fofun 6797	. . . . . 6 ⊢ (bday:No–onto→On → Fun bday)
3	1, 2	ax-mp 5	. . . . 5 ⊢ Fun bday
4		funimaexg 6626	. . . . 5 ⊢ ((Fun bday ∧ 𝐴 ∈ 𝑉) → (bday “ 𝐴) ∈ V)
5	3, 4	mpan 703	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (bday “ 𝐴) ∈ V)
6	5	uniexd 7759	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ (bday “ 𝐴) ∈ V)
7		imassrn 6197	. . . . 5 ⊢ (bday “ 𝐴) ⊆ ran bday
8		forn 6799	. . . . . 6 ⊢ (bday:No–onto→On → ran bday = On)
9	1, 8	ax-mp 5	. . . . 5 ⊢ ran bday = On
10	7, 9	sseqtri 3956	. . . 4 ⊢ (bday “ 𝐴) ⊆ On
11		ssorduni 7793	. . . 4 ⊢ ((bday “ 𝐴) ⊆ On → Ord ∪ (bday “ 𝐴))
12	10, 11	ax-mp 5	. . 3 ⊢ Ord ∪ (bday “ 𝐴)
13	6, 12	jctil 529	. 2 ⊢ (𝐴 ∈ 𝑉 → (Ord ∪ (bday “ 𝐴) ∧ ∪ (bday “ 𝐴) ∈ V))
14		elon2 6373	. . 3 ⊢ (∪ (bday “ 𝐴) ∈ On ↔ (Ord ∪ (bday “ 𝐴) ∧ ∪ (bday “ 𝐴) ∈ V))
15		onsucb 7828	. . 3 ⊢ (∪ (bday “ 𝐴) ∈ On ↔ suc ∪ (bday “ 𝐴) ∈ On)
16	14, 15	bitr3i 280	. 2 ⊢ ((Ord ∪ (bday “ 𝐴) ∧ ∪ (bday “ 𝐴) ∈ V) ↔ suc ∪ (bday “ 𝐴) ∈ On)
17	13, 16	sylib 221	1 ⊢ (𝐴 ∈ 𝑉 → suc ∪ (bday “ 𝐴) ∈ On)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895  ∪ cuni 4866  ran crn 5651   “ cima 5653  Ord word 6361  Oncon0 6362  suc csuc 6364  Fun wfun 6532  –onto→wfo 6536  Nocsur 27997  bdaycbday 27999

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-pow 5326  ax-pr 5390  ax-un 7751

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-rab 3413  df-v 3452  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-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-ord 6365  df-on 6366  df-suc 6368  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-1o 8476  df-no 28000  df-bday 28002

This theorem is used by:  noetasuplem1  28090  noetainflem1  28094