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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 35794 | . 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) | |
2 | wsuccl.1 | . . . 4 ⊢ (𝜑 → 𝑅 We 𝐴) | |
3 | weso 5680 | . . . 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 35807 | . . 3 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 (∀𝑏 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑐𝑅𝑏))) |
9 | 4, 8 | infcl 9526 | . 2 ⊢ (𝜑 → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ 𝐴) |
10 | 1, 9 | eqeltrid 2843 | 1 ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∃wrex 3068 class class class wbr 5148 Or wor 5596 Se wse 5639 We wwe 5640 ◡ccnv 5688 Predcpred 6322 infcinf 9479 wsuccwsuc 35792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-iota 6516 df-riota 7388 df-sup 9480 df-inf 9481 df-wsuc 35794 |
This theorem is referenced by: (None) |
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