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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 35408 | . 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) | |
2 | wsuccl.1 | . . . 4 ⊢ (𝜑 → 𝑅 We 𝐴) | |
3 | weso 5669 | . . . 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 35421 | . . 3 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 (∀𝑏 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑐𝑅𝑏))) |
9 | 4, 8 | infcl 9512 | . 2 ⊢ (𝜑 → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ 𝐴) |
10 | 1, 9 | eqeltrid 2833 | 1 ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ∃wrex 3067 class class class wbr 5148 Or wor 5589 Se wse 5631 We wwe 5632 ◡ccnv 5677 Predcpred 6304 infcinf 9465 wsuccwsuc 35406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-iota 6500 df-riota 7376 df-sup 9466 df-inf 9467 df-wsuc 35408 |
This theorem is referenced by: (None) |
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