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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 5882 | . . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝑉) → (𝑌◡𝑅𝑋 ↔ 𝑋𝑅𝑌)) | |
5 | 2, 3, 4 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → (𝑌◡𝑅𝑋 ↔ 𝑋𝑅𝑌)) |
6 | 1, 5 | mpbird 256 | . . . 4 ⊢ (𝜑 → 𝑌◡𝑅𝑋) |
7 | elpredg 6321 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋) ↔ 𝑌◡𝑅𝑋)) | |
8 | 3, 2, 7 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋) ↔ 𝑌◡𝑅𝑋)) |
9 | 6, 8 | mpbird 256 | . . 3 ⊢ (𝜑 → 𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋)) |
10 | wsuclb.1 | . . . . 5 ⊢ (𝜑 → 𝑅 We 𝐴) | |
11 | weso 5669 | . . . . 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 5153 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋𝑅𝑦 ↔ 𝑋𝑅𝑌)) | |
15 | 14 | rspcev 3606 | . . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑋𝑅𝑌) → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦) |
16 | 2, 1, 15 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦) |
17 | 10, 13, 3, 16 | wsuclem 35552 | . . . 4 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 (∀𝑏 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑐𝑅𝑏))) |
18 | 12, 17 | inflb 9514 | . . 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 35539 | . . 3 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) | |
21 | 20 | breq2i 5157 | . 2 ⊢ (𝑌𝑅wsuc(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)) |
22 | 19, 21 | sylnibr 328 | 1 ⊢ (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∈ wcel 2098 ∃wrex 3059 class class class wbr 5149 Or wor 5589 Se wse 5631 We wwe 5632 ◡ccnv 5677 Predcpred 6306 infcinf 9466 wsuccwsuc 35537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-iota 6501 df-riota 7375 df-sup 9467 df-inf 9468 df-wsuc 35539 |
This theorem is referenced by: (None) |
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