![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > wsuclb | Structured version Visualization version GIF version |
Description: A well-founded successor is a lower bound on points after 𝑋. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
Ref | Expression |
---|---|
wsuclb.1 | ⊢ (𝜑 → 𝑅 We 𝐴) |
wsuclb.2 | ⊢ (𝜑 → 𝑅 Se 𝐴) |
wsuclb.3 | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
wsuclb.4 | ⊢ (𝜑 → 𝑌 ∈ 𝐴) |
wsuclb.5 | ⊢ (𝜑 → 𝑋𝑅𝑌) |
Ref | Expression |
---|---|
wsuclb | ⊢ (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wsuclb.5 | . . . . 5 ⊢ (𝜑 → 𝑋𝑅𝑌) | |
2 | wsuclb.4 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐴) | |
3 | wsuclb.3 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
4 | brcnvg 5547 | . . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝑉) → (𝑌◡𝑅𝑋 ↔ 𝑋𝑅𝑌)) | |
5 | 2, 3, 4 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → (𝑌◡𝑅𝑋 ↔ 𝑋𝑅𝑌)) |
6 | 1, 5 | mpbird 249 | . . . 4 ⊢ (𝜑 → 𝑌◡𝑅𝑋) |
7 | elpredg 5947 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋) ↔ 𝑌◡𝑅𝑋)) | |
8 | 3, 2, 7 | syl2anc 579 | . . . 4 ⊢ (𝜑 → (𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋) ↔ 𝑌◡𝑅𝑋)) |
9 | 6, 8 | mpbird 249 | . . 3 ⊢ (𝜑 → 𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋)) |
10 | wsuclb.1 | . . . . 5 ⊢ (𝜑 → 𝑅 We 𝐴) | |
11 | weso 5346 | . . . . 5 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴) |
13 | wsuclb.2 | . . . . 5 ⊢ (𝜑 → 𝑅 Se 𝐴) | |
14 | breq2 4890 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋𝑅𝑦 ↔ 𝑋𝑅𝑌)) | |
15 | 14 | rspcev 3511 | . . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑋𝑅𝑌) → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦) |
16 | 2, 1, 15 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦) |
17 | 10, 13, 3, 16 | wsuclem 32359 | . . . 4 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 (∀𝑏 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑐𝑅𝑏))) |
18 | 12, 17 | inflb 8683 | . . 3 ⊢ (𝜑 → (𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋) → ¬ 𝑌𝑅inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅))) |
19 | 9, 18 | mpd 15 | . 2 ⊢ (𝜑 → ¬ 𝑌𝑅inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)) |
20 | df-wsuc 32346 | . . 3 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) | |
21 | 20 | breq2i 4894 | . 2 ⊢ (𝑌𝑅wsuc(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)) |
22 | 19, 21 | sylnibr 321 | 1 ⊢ (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∈ wcel 2107 ∃wrex 3091 class class class wbr 4886 Or wor 5273 Se wse 5312 We wwe 5313 ◡ccnv 5354 Predcpred 5932 infcinf 8635 wsuccwsuc 32344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-cnv 5363 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-iota 6099 df-riota 6883 df-sup 8636 df-inf 8637 df-wsuc 32346 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |