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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 34772 | . 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) | |
2 | wsuccl.1 | . . . 4 ⊢ (𝜑 → 𝑅 We 𝐴) | |
3 | weso 5666 | . . . 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 34785 | . . 3 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 (∀𝑏 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑐𝑅𝑏))) |
9 | 4, 8 | infcl 9479 | . 2 ⊢ (𝜑 → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ 𝐴) |
10 | 1, 9 | eqeltrid 2837 | 1 ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∃wrex 3070 class class class wbr 5147 Or wor 5586 Se wse 5628 We wwe 5629 ◡ccnv 5674 Predcpred 6296 infcinf 9432 wsuccwsuc 34770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-cnv 5683 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-iota 6492 df-riota 7361 df-sup 9433 df-inf 9434 df-wsuc 34772 |
This theorem is referenced by: (None) |
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