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Mirrors > Home > MPE Home > Th. List > etasslt2 | Structured version Visualization version GIF version |
Description: A version of etasslt 27174 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 27134 | . . . . . 6 ⊢ Fun bday | |
2 | ssltex1 27148 | . . . . . . 7 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V) | |
3 | ssltex2 27149 | . . . . . . 7 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V) | |
4 | unexg 7684 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V) | |
5 | 2, 3, 4 | syl2anc 585 | . . . . . 6 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∪ 𝐵) ∈ V) |
6 | funimaexg 6588 | . . . . . 6 ⊢ ((Fun bday ∧ (𝐴 ∪ 𝐵) ∈ V) → ( bday “ (𝐴 ∪ 𝐵)) ∈ V) | |
7 | 1, 5, 6 | sylancr 588 | . . . . 5 ⊢ (𝐴 <<s 𝐵 → ( bday “ (𝐴 ∪ 𝐵)) ∈ V) |
8 | 7 | uniexd 7680 | . . . 4 ⊢ (𝐴 <<s 𝐵 → ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V) |
9 | imassrn 6025 | . . . . . . 7 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ ran bday | |
10 | bdayrn 27137 | . . . . . . 7 ⊢ ran bday = On | |
11 | 9, 10 | sseqtri 3981 | . . . . . 6 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ On |
12 | ssorduni 7714 | . . . . . 6 ⊢ (( bday “ (𝐴 ∪ 𝐵)) ⊆ On → Ord ∪ ( bday “ (𝐴 ∪ 𝐵))) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ Ord ∪ ( bday “ (𝐴 ∪ 𝐵)) |
14 | elon2 6329 | . . . . 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 233 | . . 3 ⊢ (𝐴 <<s 𝐵 → ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On) |
17 | onsucb 7753 | . . 3 ⊢ (∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ suc ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On) | |
18 | 16, 17 | sylib 217 | . 2 ⊢ (𝐴 <<s 𝐵 → suc ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On) |
19 | onsucuni 7764 | . . 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 27174 | . 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 1464 | 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 1088 ∈ wcel 2107 ∃wrex 3070 Vcvv 3444 ∪ cun 3909 ⊆ wss 3911 {csn 4587 ∪ cuni 4866 class class class wbr 5106 ran crn 5635 “ cima 5637 Ord word 6317 Oncon0 6318 suc csuc 6320 Fun wfun 6491 ‘cfv 6497 No csur 27004 bday cbday 27006 <<s csslt 27142 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-1o 8413 df-2o 8414 df-no 27007 df-slt 27008 df-bday 27009 df-sslt 27143 |
This theorem is referenced by: scutbdaybnd2 27177 |
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