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Mirrors > Home > MPE Home > Th. List > 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 27109 | . . . . . 6 ⊢ bday : No –onto→On | |
2 | fofun 6794 | . . . . . 6 ⊢ ( bday : No –onto→On → Fun bday ) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ Fun bday |
4 | funimaexg 6624 | . . . . 5 ⊢ ((Fun bday ∧ 𝐴 ∈ 𝑉) → ( bday “ 𝐴) ∈ V) | |
5 | 3, 4 | mpan 688 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ( bday “ 𝐴) ∈ V) |
6 | 5 | uniexd 7716 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ ( bday “ 𝐴) ∈ V) |
7 | imassrn 6061 | . . . . 5 ⊢ ( bday “ 𝐴) ⊆ ran bday | |
8 | forn 6796 | . . . . . 6 ⊢ ( bday : No –onto→On → ran bday = On) | |
9 | 1, 8 | ax-mp 5 | . . . . 5 ⊢ ran bday = On |
10 | 7, 9 | sseqtri 4015 | . . . 4 ⊢ ( bday “ 𝐴) ⊆ On |
11 | ssorduni 7750 | . . . 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 6365 | . . 3 ⊢ (∪ ( bday “ 𝐴) ∈ On ↔ (Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V)) | |
15 | onsucb 7789 | . . 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 1541 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3945 ∪ cuni 4902 ran crn 5671 “ cima 5673 Ord word 6353 Oncon0 6354 suc csuc 6356 Fun wfun 6527 –onto→wfo 6531 No csur 27072 bday cbday 27074 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pr 5421 ax-un 7709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-ord 6357 df-on 6358 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-1o 8450 df-no 27075 df-bday 27077 |
This theorem is referenced by: noetasuplem1 27165 noetainflem1 27169 |
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