MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bdayimaon Structured version   Visualization version   GIF version

Theorem bdayimaon 27064
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 27048 . . . . . 6 bday : No –ontoβ†’On
2 fofun 6761 . . . . . 6 ( bday : No –ontoβ†’On β†’ Fun bday )
31, 2ax-mp 5 . . . . 5 Fun bday
4 funimaexg 6591 . . . . 5 ((Fun bday ∧ 𝐴 ∈ 𝑉) β†’ ( bday β€œ 𝐴) ∈ V)
53, 4mpan 689 . . . 4 (𝐴 ∈ 𝑉 β†’ ( bday β€œ 𝐴) ∈ V)
65uniexd 7683 . . 3 (𝐴 ∈ 𝑉 β†’ βˆͺ ( bday β€œ 𝐴) ∈ V)
7 imassrn 6028 . . . . 5 ( bday β€œ 𝐴) βŠ† ran bday
8 forn 6763 . . . . . 6 ( bday : No –ontoβ†’On β†’ ran bday = On)
91, 8ax-mp 5 . . . . 5 ran bday = On
107, 9sseqtri 3984 . . . 4 ( bday β€œ 𝐴) βŠ† On
11 ssorduni 7717 . . . 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 521 . 2 (𝐴 ∈ 𝑉 β†’ (Ord βˆͺ ( bday β€œ 𝐴) ∧ βˆͺ ( bday β€œ 𝐴) ∈ V))
14 elon2 6332 . . 3 (βˆͺ ( bday β€œ 𝐴) ∈ On ↔ (Ord βˆͺ ( bday β€œ 𝐴) ∧ βˆͺ ( bday β€œ 𝐴) ∈ V))
15 onsucb 7756 . . 3 (βˆͺ ( bday β€œ 𝐴) ∈ On ↔ suc βˆͺ ( bday β€œ 𝐴) ∈ On)
1614, 15bitr3i 277 . 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 397   = wceq 1542   ∈ wcel 2107  Vcvv 3447   βŠ† wss 3914  βˆͺ cuni 4869  ran crn 5638   β€œ cima 5640  Ord word 6320  Oncon0 6321  suc csuc 6323  Fun wfun 6494  β€“ontoβ†’wfo 6498   No csur 27011   bday cbday 27013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-1o 8416  df-no 27014  df-bday 27016
This theorem is referenced by:  noetasuplem1  27104  noetainflem1  27108
  Copyright terms: Public domain W3C validator