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Theorem bdayimaon 27654
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 β†’ suc βˆͺ ( bday β€œ 𝐴) ∈ On)

Proof of Theorem bdayimaon
StepHypRef Expression
1 bdayfo 27638 . . . . . 6 bday : No –ontoβ†’On
2 fofun 6817 . . . . . 6 ( bday : No –ontoβ†’On β†’ Fun bday )
31, 2ax-mp 5 . . . . 5 Fun bday
4 funimaexg 6644 . . . . 5 ((Fun bday ∧ 𝐴 ∈ 𝑉) β†’ ( bday β€œ 𝐴) ∈ V)
53, 4mpan 688 . . . 4 (𝐴 ∈ 𝑉 β†’ ( bday β€œ 𝐴) ∈ V)
65uniexd 7755 . . 3 (𝐴 ∈ 𝑉 β†’ βˆͺ ( bday β€œ 𝐴) ∈ V)
7 imassrn 6079 . . . . 5 ( bday β€œ 𝐴) βŠ† ran bday
8 forn 6819 . . . . . 6 ( bday : No –ontoβ†’On β†’ ran bday = On)
91, 8ax-mp 5 . . . . 5 ran bday = On
107, 9sseqtri 4018 . . . 4 ( bday β€œ 𝐴) βŠ† On
11 ssorduni 7789 . . . 4 (( bday β€œ 𝐴) βŠ† On β†’ Ord βˆͺ ( bday β€œ 𝐴))
1210, 11ax-mp 5 . . 3 Ord βˆͺ ( bday β€œ 𝐴)
136, 12jctil 518 . 2 (𝐴 ∈ 𝑉 β†’ (Ord βˆͺ ( bday β€œ 𝐴) ∧ βˆͺ ( bday β€œ 𝐴) ∈ V))
14 elon2 6385 . . 3 (βˆͺ ( bday β€œ 𝐴) ∈ On ↔ (Ord βˆͺ ( bday β€œ 𝐴) ∧ βˆͺ ( bday β€œ 𝐴) ∈ V))
15 onsucb 7828 . . 3 (βˆͺ ( bday β€œ 𝐴) ∈ On ↔ suc βˆͺ ( bday β€œ 𝐴) ∈ On)
1614, 15bitr3i 276 . 2 ((Ord βˆͺ ( bday β€œ 𝐴) ∧ βˆͺ ( bday β€œ 𝐴) ∈ V) ↔ suc βˆͺ ( bday β€œ 𝐴) ∈ On)
1713, 16sylib 217 1 (𝐴 ∈ 𝑉 β†’ suc βˆͺ ( bday β€œ 𝐴) ∈ On)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3473   βŠ† wss 3949  βˆͺ cuni 4912  ran crn 5683   β€œ cima 5685  Ord word 6373  Oncon0 6374  suc csuc 6376  Fun wfun 6547  β€“ontoβ†’wfo 6551   No csur 27601   bday cbday 27603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-1o 8495  df-no 27604  df-bday 27606
This theorem is referenced by:  noetasuplem1  27694  noetainflem1  27698
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