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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 33880 | . . . . . 6 ⊢ bday : No –onto→On | |
2 | fofun 6689 | . . . . . 6 ⊢ ( bday : No –onto→On → Fun bday ) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ Fun bday |
4 | funimaexg 6520 | . . . . 5 ⊢ ((Fun bday ∧ 𝐴 ∈ 𝑉) → ( bday “ 𝐴) ∈ V) | |
5 | 3, 4 | mpan 687 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ( bday “ 𝐴) ∈ V) |
6 | 5 | uniexd 7595 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ ( bday “ 𝐴) ∈ V) |
7 | imassrn 5980 | . . . . 5 ⊢ ( bday “ 𝐴) ⊆ ran bday | |
8 | forn 6691 | . . . . . 6 ⊢ ( bday : No –onto→On → ran bday = On) | |
9 | 1, 8 | ax-mp 5 | . . . . 5 ⊢ ran bday = On |
10 | 7, 9 | sseqtri 3957 | . . . 4 ⊢ ( bday “ 𝐴) ⊆ On |
11 | ssorduni 7629 | . . . 4 ⊢ (( bday “ 𝐴) ⊆ On → Ord ∪ ( bday “ 𝐴)) | |
12 | 10, 11 | ax-mp 5 | . . 3 ⊢ Ord ∪ ( bday “ 𝐴) |
13 | 6, 12 | jctil 520 | . 2 ⊢ (𝐴 ∈ 𝑉 → (Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V)) |
14 | elon2 6277 | . . 3 ⊢ (∪ ( bday “ 𝐴) ∈ On ↔ (Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V)) | |
15 | sucelon 7664 | . . 3 ⊢ (∪ ( bday “ 𝐴) ∈ On ↔ suc ∪ ( bday “ 𝐴) ∈ On) | |
16 | 14, 15 | bitr3i 276 | . 2 ⊢ ((Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V) ↔ suc ∪ ( bday “ 𝐴) ∈ On) |
17 | 13, 16 | sylib 217 | 1 ⊢ (𝐴 ∈ 𝑉 → suc ∪ ( bday “ 𝐴) ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ⊆ wss 3887 ∪ cuni 4839 ran crn 5590 “ cima 5592 Ord word 6265 Oncon0 6266 suc csuc 6268 Fun wfun 6427 –onto→wfo 6431 No csur 33843 bday cbday 33845 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-ord 6269 df-on 6270 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-1o 8297 df-no 33846 df-bday 33848 |
This theorem is referenced by: noetasuplem1 33936 noetainflem1 33940 |
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