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Mirrors > Home > MPE Home > Th. List > Mathboxes > etasslt2 | Structured version Visualization version GIF version |
Description: A version of etasslt 33594 with fewer hypotheses but a weaker upper bound. (Contributed by Scott Fenton, 10-Dec-2021.) |
Ref | Expression |
---|---|
etasslt2 | ⊢ (𝐴 <<s 𝐵 → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdayfun 33556 | . . . . . 6 ⊢ Fun bday | |
2 | ssltex1 33570 | . . . . . . 7 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V) | |
3 | ssltex2 33571 | . . . . . . 7 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V) | |
4 | unexg 7475 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V) | |
5 | 2, 3, 4 | syl2anc 587 | . . . . . 6 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∪ 𝐵) ∈ V) |
6 | funimaexg 6425 | . . . . . 6 ⊢ ((Fun bday ∧ (𝐴 ∪ 𝐵) ∈ V) → ( bday “ (𝐴 ∪ 𝐵)) ∈ V) | |
7 | 1, 5, 6 | sylancr 590 | . . . . 5 ⊢ (𝐴 <<s 𝐵 → ( bday “ (𝐴 ∪ 𝐵)) ∈ V) |
8 | 7 | uniexd 7471 | . . . 4 ⊢ (𝐴 <<s 𝐵 → ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V) |
9 | imassrn 5916 | . . . . . . 7 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ ran bday | |
10 | bdayrn 33559 | . . . . . . 7 ⊢ ran bday = On | |
11 | 9, 10 | sseqtri 3930 | . . . . . 6 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ On |
12 | ssorduni 7504 | . . . . . 6 ⊢ (( bday “ (𝐴 ∪ 𝐵)) ⊆ On → Ord ∪ ( bday “ (𝐴 ∪ 𝐵))) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ Ord ∪ ( bday “ (𝐴 ∪ 𝐵)) |
14 | elon2 6184 | . . . . 5 ⊢ (∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ (Ord ∪ ( bday “ (𝐴 ∪ 𝐵)) ∧ ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V)) | |
15 | 13, 14 | mpbiran 708 | . . . 4 ⊢ (∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V) |
16 | 8, 15 | sylibr 237 | . . 3 ⊢ (𝐴 <<s 𝐵 → ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On) |
17 | sucelon 7536 | . . 3 ⊢ (∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ suc ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On) | |
18 | 16, 17 | sylib 221 | . 2 ⊢ (𝐴 <<s 𝐵 → suc ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On) |
19 | onsucuni 7547 | . . 3 ⊢ (( bday “ (𝐴 ∪ 𝐵)) ⊆ On → ( bday “ (𝐴 ∪ 𝐵)) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))) | |
20 | 11, 19 | mp1i 13 | . 2 ⊢ (𝐴 <<s 𝐵 → ( bday “ (𝐴 ∪ 𝐵)) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))) |
21 | etasslt 33594 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ suc ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))) → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))) | |
22 | 18, 20, 21 | mpd3an23 1460 | 1 ⊢ (𝐴 <<s 𝐵 → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 ∈ wcel 2111 ∃wrex 3071 Vcvv 3409 ∪ cun 3858 ⊆ wss 3860 {csn 4525 ∪ cuni 4801 class class class wbr 5035 ran crn 5528 “ cima 5530 Ord word 6172 Oncon0 6173 suc csuc 6175 Fun wfun 6333 ‘cfv 6339 No csur 33432 bday cbday 33434 <<s csslt 33564 |
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 2729 ax-rep 5159 ax-sep 5172 ax-nul 5179 ax-pr 5301 ax-un 7464 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-br 5036 df-opab 5098 df-mpt 5116 df-tr 5142 df-id 5433 df-eprel 5438 df-po 5446 df-so 5447 df-fr 5486 df-we 5488 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-ord 6176 df-on 6177 df-suc 6179 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7113 df-1o 8117 df-2o 8118 df-no 33435 df-slt 33436 df-bday 33437 df-sslt 33565 |
This theorem is referenced by: scutbdaybnd2 33597 |
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