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Mirrors > Home > MPE Home > Th. List > etasslt2 | Structured version Visualization version GIF version |
Description: A version of etasslt 27873 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 27832 | . . . . . 6 ⊢ Fun bday | |
2 | ssltex1 27846 | . . . . . . 7 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V) | |
3 | ssltex2 27847 | . . . . . . 7 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V) | |
4 | unexg 7762 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V) | |
5 | 2, 3, 4 | syl2anc 584 | . . . . . 6 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∪ 𝐵) ∈ V) |
6 | funimaexg 6654 | . . . . . 6 ⊢ ((Fun bday ∧ (𝐴 ∪ 𝐵) ∈ V) → ( bday “ (𝐴 ∪ 𝐵)) ∈ V) | |
7 | 1, 5, 6 | sylancr 587 | . . . . 5 ⊢ (𝐴 <<s 𝐵 → ( bday “ (𝐴 ∪ 𝐵)) ∈ V) |
8 | 7 | uniexd 7761 | . . . 4 ⊢ (𝐴 <<s 𝐵 → ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V) |
9 | imassrn 6091 | . . . . . . 7 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ ran bday | |
10 | bdayrn 27835 | . . . . . . 7 ⊢ ran bday = On | |
11 | 9, 10 | sseqtri 4032 | . . . . . 6 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ On |
12 | ssorduni 7798 | . . . . . 6 ⊢ (( bday “ (𝐴 ∪ 𝐵)) ⊆ On → Ord ∪ ( bday “ (𝐴 ∪ 𝐵))) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ Ord ∪ ( bday “ (𝐴 ∪ 𝐵)) |
14 | elon2 6397 | . . . . 5 ⊢ (∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ (Ord ∪ ( bday “ (𝐴 ∪ 𝐵)) ∧ ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V)) | |
15 | 13, 14 | mpbiran 709 | . . . 4 ⊢ (∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V) |
16 | 8, 15 | sylibr 234 | . . 3 ⊢ (𝐴 <<s 𝐵 → ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On) |
17 | onsucb 7837 | . . 3 ⊢ (∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ suc ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On) | |
18 | 16, 17 | sylib 218 | . 2 ⊢ (𝐴 <<s 𝐵 → suc ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On) |
19 | onsucuni 7848 | . . 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 27873 | . 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 1462 | 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 1086 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 ∪ cun 3961 ⊆ wss 3963 {csn 4631 ∪ cuni 4912 class class class wbr 5148 ran crn 5690 “ cima 5692 Ord word 6385 Oncon0 6386 suc csuc 6388 Fun wfun 6557 ‘cfv 6563 No csur 27699 bday cbday 27701 <<s csslt 27840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-1o 8505 df-2o 8506 df-no 27702 df-slt 27703 df-bday 27704 df-sslt 27841 |
This theorem is referenced by: scutbdaybnd2 27876 |
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