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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 5335 | . . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝑉) → (𝑌◡𝑅𝑋 ↔ 𝑋𝑅𝑌)) | |
5 | 2, 3, 4 | syl2anc 694 | . . . . 5 ⊢ (𝜑 → (𝑌◡𝑅𝑋 ↔ 𝑋𝑅𝑌)) |
6 | 1, 5 | mpbird 247 | . . . 4 ⊢ (𝜑 → 𝑌◡𝑅𝑋) |
7 | elpredg 5732 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋) ↔ 𝑌◡𝑅𝑋)) | |
8 | 3, 2, 7 | syl2anc 694 | . . . 4 ⊢ (𝜑 → (𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋) ↔ 𝑌◡𝑅𝑋)) |
9 | 6, 8 | mpbird 247 | . . 3 ⊢ (𝜑 → 𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋)) |
10 | wsuclb.1 | . . . . 5 ⊢ (𝜑 → 𝑅 We 𝐴) | |
11 | weso 5134 | . . . . 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 4689 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋𝑅𝑦 ↔ 𝑋𝑅𝑌)) | |
15 | 14 | rspcev 3340 | . . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑋𝑅𝑌) → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦) |
16 | 2, 1, 15 | syl2anc 694 | . . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦) |
17 | 10, 13, 3, 16 | wsuclem 31895 | . . . 4 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 (∀𝑏 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑐𝑅𝑏))) |
18 | 12, 17 | inflb 8436 | . . 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 31882 | . . 3 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) | |
21 | 20 | breq2i 4693 | . 2 ⊢ (𝑌𝑅wsuc(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)) |
22 | 19, 21 | sylnibr 318 | 1 ⊢ (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∈ wcel 2030 ∃wrex 2942 class class class wbr 4685 Or wor 5063 Se wse 5100 We wwe 5101 ◡ccnv 5142 Predcpred 5717 infcinf 8388 wsuccwsuc 31880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-cnv 5151 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-iota 5889 df-riota 6651 df-sup 8389 df-inf 8390 df-wsuc 31882 |
This theorem is referenced by: (None) |
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