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Theorem nobndlem1 30937
Description: Lemma for nobndup 30945 and nobnddown 30946. 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
nobndlem1 (𝐴𝑉 → suc ( bday 𝐴) ∈ On)

Proof of Theorem nobndlem1
StepHypRef Expression
1 bdayfun 30921 . . . . 5 Fun bday
2 funimaexg 5774 . . . . 5 ((Fun bday 𝐴𝑉) → ( bday 𝐴) ∈ V)
31, 2mpan 701 . . . 4 (𝐴𝑉 → ( bday 𝐴) ∈ V)
4 uniexg 6727 . . . 4 (( bday 𝐴) ∈ V → ( bday 𝐴) ∈ V)
53, 4syl 17 . . 3 (𝐴𝑉 ( bday 𝐴) ∈ V)
6 imassrn 5286 . . . . 5 ( bday 𝐴) ⊆ ran bday
7 bdayrn 30922 . . . . 5 ran bday = On
86, 7sseqtri 3504 . . . 4 ( bday 𝐴) ⊆ On
9 ssorduni 6751 . . . 4 (( bday 𝐴) ⊆ On → Ord ( bday 𝐴))
108, 9ax-mp 5 . . 3 Ord ( bday 𝐴)
115, 10jctil 557 . 2 (𝐴𝑉 → (Ord ( bday 𝐴) ∧ ( bday 𝐴) ∈ V))
12 elon2 5541 . . 3 ( ( bday 𝐴) ∈ On ↔ (Ord ( bday 𝐴) ∧ ( bday 𝐴) ∈ V))
13 sucelon 6783 . . 3 ( ( bday 𝐴) ∈ On ↔ suc ( bday 𝐴) ∈ On)
1412, 13bitr3i 264 . 2 ((Ord ( bday 𝐴) ∧ ( bday 𝐴) ∈ V) ↔ suc ( bday 𝐴) ∈ On)
1511, 14sylib 206 1 (𝐴𝑉 → suc ( bday 𝐴) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 1938  Vcvv 3077  wss 3444   cuni 4270  ran crn 4933  cima 4935  Ord word 5529  Oncon0 5530  suc csuc 5532  Fun wfun 5683   bday cbday 30885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pr 4732  ax-un 6721
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-ord 5533  df-on 5534  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-1o 7321  df-no 30886  df-bday 30888
This theorem is referenced by:  nobndlem2  30938  nobndlem8  30944  nofulllem4  30950
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