![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > wsuccl | Structured version Visualization version GIF version |
Description: If 𝑋 is a set with an 𝑅 successor in 𝐴, then its well-founded successor is a member of 𝐴. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
Ref | Expression |
---|---|
wsuccl.1 | ⊢ (𝜑 → 𝑅 We 𝐴) |
wsuccl.2 | ⊢ (𝜑 → 𝑅 Se 𝐴) |
wsuccl.3 | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
wsuccl.4 | ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦) |
Ref | Expression |
---|---|
wsuccl | ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-wsuc 35307 | . 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) | |
2 | wsuccl.1 | . . . 4 ⊢ (𝜑 → 𝑅 We 𝐴) | |
3 | weso 5658 | . . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 Or 𝐴) |
5 | wsuccl.2 | . . . 4 ⊢ (𝜑 → 𝑅 Se 𝐴) | |
6 | wsuccl.3 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
7 | wsuccl.4 | . . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦) | |
8 | 2, 5, 6, 7 | wsuclem 35320 | . . 3 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 (∀𝑏 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑐𝑅𝑏))) |
9 | 4, 8 | infcl 9480 | . 2 ⊢ (𝜑 → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ 𝐴) |
10 | 1, 9 | eqeltrid 2829 | 1 ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ∃wrex 3062 class class class wbr 5139 Or wor 5578 Se wse 5620 We wwe 5621 ◡ccnv 5666 Predcpred 6290 infcinf 9433 wsuccwsuc 35305 |
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 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-cnv 5675 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-iota 6486 df-riota 7358 df-sup 9434 df-inf 9435 df-wsuc 35307 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |