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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 5890 | . . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝑉) → (𝑌◡𝑅𝑋 ↔ 𝑋𝑅𝑌)) | |
| 5 | 2, 3, 4 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑌◡𝑅𝑋 ↔ 𝑋𝑅𝑌)) |
| 6 | 1, 5 | mpbird 257 | . . . 4 ⊢ (𝜑 → 𝑌◡𝑅𝑋) |
| 7 | elpredg 6335 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋) ↔ 𝑌◡𝑅𝑋)) | |
| 8 | 3, 2, 7 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋) ↔ 𝑌◡𝑅𝑋)) |
| 9 | 6, 8 | mpbird 257 | . . 3 ⊢ (𝜑 → 𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋)) |
| 10 | wsuclb.1 | . . . . 5 ⊢ (𝜑 → 𝑅 We 𝐴) | |
| 11 | weso 5676 | . . . . 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 5147 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋𝑅𝑦 ↔ 𝑋𝑅𝑌)) | |
| 15 | 14 | rspcev 3622 | . . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑋𝑅𝑌) → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦) |
| 16 | 2, 1, 15 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦) |
| 17 | 10, 13, 3, 16 | wsuclem 35826 | . . . 4 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 (∀𝑏 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑐𝑅𝑏))) |
| 18 | 12, 17 | inflb 9529 | . . 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 35813 | . . 3 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) | |
| 21 | 20 | breq2i 5151 | . 2 ⊢ (𝑌𝑅wsuc(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)) |
| 22 | 19, 21 | sylnibr 329 | 1 ⊢ (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∈ wcel 2108 ∃wrex 3070 class class class wbr 5143 Or wor 5591 Se wse 5635 We wwe 5636 ◡ccnv 5684 Predcpred 6320 infcinf 9481 wsuccwsuc 35811 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-riota 7388 df-sup 9482 df-inf 9483 df-wsuc 35813 |
| This theorem is referenced by: (None) |
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