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Theorem nobndlem1 31608
Description: Lemma for nobndup 31616 and nobnddown 31617. 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 31592 . . . . 5 Fun bday
2 funimaexg 5943 . . . . 5 ((Fun bday 𝐴𝑉) → ( bday 𝐴) ∈ V)
31, 2mpan 705 . . . 4 (𝐴𝑉 → ( bday 𝐴) ∈ V)
4 uniexg 6920 . . . 4 (( bday 𝐴) ∈ V → ( bday 𝐴) ∈ V)
53, 4syl 17 . . 3 (𝐴𝑉 ( bday 𝐴) ∈ V)
6 imassrn 5446 . . . . 5 ( bday 𝐴) ⊆ ran bday
7 bdayrn 31593 . . . . 5 ran bday = On
86, 7sseqtri 3622 . . . 4 ( bday 𝐴) ⊆ On
9 ssorduni 6947 . . . 4 (( bday 𝐴) ⊆ On → Ord ( bday 𝐴))
108, 9ax-mp 5 . . 3 Ord ( bday 𝐴)
115, 10jctil 559 . 2 (𝐴𝑉 → (Ord ( bday 𝐴) ∧ ( bday 𝐴) ∈ V))
12 elon2 5703 . . 3 ( ( bday 𝐴) ∈ On ↔ (Ord ( bday 𝐴) ∧ ( bday 𝐴) ∈ V))
13 sucelon 6979 . . 3 ( ( bday 𝐴) ∈ On ↔ suc ( bday 𝐴) ∈ On)
1412, 13bitr3i 266 . 2 ((Ord ( bday 𝐴) ∧ ( bday 𝐴) ∈ V) ↔ suc ( bday 𝐴) ∈ On)
1511, 14sylib 208 1 (𝐴𝑉 → suc ( bday 𝐴) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  Vcvv 3190  wss 3560   cuni 4409  ran crn 5085  cima 5087  Ord word 5691  Oncon0 5692  suc csuc 5694  Fun wfun 5851   bday cbday 31549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-ord 5695  df-on 5696  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-1o 7520  df-no 31550  df-bday 31552
This theorem is referenced by:  nobndlem2  31609  nobndlem8  31615
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