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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 34426 | . 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) | |
2 | wsuccl.1 | . . . 4 ⊢ (𝜑 → 𝑅 We 𝐴) | |
3 | weso 5629 | . . . 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 34439 | . . 3 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 (∀𝑏 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑐𝑅𝑏))) |
9 | 4, 8 | infcl 9431 | . 2 ⊢ (𝜑 → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ 𝐴) |
10 | 1, 9 | eqeltrid 2842 | 1 ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∃wrex 3074 class class class wbr 5110 Or wor 5549 Se wse 5591 We wwe 5592 ◡ccnv 5637 Predcpred 6257 infcinf 9384 wsuccwsuc 34424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-cnv 5646 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-iota 6453 df-riota 7318 df-sup 9385 df-inf 9386 df-wsuc 34426 |
This theorem is referenced by: (None) |
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