![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bdayimaon | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
bdayimaon | ⊢ (𝐴 ∈ 𝑉 → suc ∪ ( bday “ 𝐴) ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdayfo 33295 | . . . . . 6 ⊢ bday : No –onto→On | |
2 | fofun 6566 | . . . . . 6 ⊢ ( bday : No –onto→On → Fun bday ) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ Fun bday |
4 | funimaexg 6410 | . . . . 5 ⊢ ((Fun bday ∧ 𝐴 ∈ 𝑉) → ( bday “ 𝐴) ∈ V) | |
5 | 3, 4 | mpan 689 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ( bday “ 𝐴) ∈ V) |
6 | 5 | uniexd 7448 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ ( bday “ 𝐴) ∈ V) |
7 | imassrn 5907 | . . . . 5 ⊢ ( bday “ 𝐴) ⊆ ran bday | |
8 | forn 6568 | . . . . . 6 ⊢ ( bday : No –onto→On → ran bday = On) | |
9 | 1, 8 | ax-mp 5 | . . . . 5 ⊢ ran bday = On |
10 | 7, 9 | sseqtri 3951 | . . . 4 ⊢ ( bday “ 𝐴) ⊆ On |
11 | ssorduni 7480 | . . . 4 ⊢ (( bday “ 𝐴) ⊆ On → Ord ∪ ( bday “ 𝐴)) | |
12 | 10, 11 | ax-mp 5 | . . 3 ⊢ Ord ∪ ( bday “ 𝐴) |
13 | 6, 12 | jctil 523 | . 2 ⊢ (𝐴 ∈ 𝑉 → (Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V)) |
14 | elon2 6170 | . . 3 ⊢ (∪ ( bday “ 𝐴) ∈ On ↔ (Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V)) | |
15 | sucelon 7512 | . . 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) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 ∪ cuni 4800 ran crn 5520 “ cima 5522 Ord word 6158 Oncon0 6159 suc csuc 6161 Fun wfun 6318 –onto→wfo 6322 No csur 33260 bday cbday 33262 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-1o 8085 df-no 33263 df-bday 33265 |
This theorem is referenced by: noetalem1 33330 |
Copyright terms: Public domain | W3C validator |