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Mirrors > Home > MPE Home > Th. List > Mathboxes > wsucex | Structured version Visualization version GIF version |
Description: Existence theorem for well-founded successor. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
Ref | Expression |
---|---|
wsucex.1 | ⊢ (𝜑 → 𝑅 Or 𝐴) |
Ref | Expression |
---|---|
wsucex | ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-wsuc 33806 | . 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) | |
2 | wsucex.1 | . . 3 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
3 | 2 | infexd 9242 | . 2 ⊢ (𝜑 → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ V) |
4 | 1, 3 | eqeltrid 2843 | 1 ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3432 Or wor 5502 ◡ccnv 5588 Predcpred 6201 infcinf 9200 wsuccwsuc 33804 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-po 5503 df-so 5504 df-cnv 5597 df-sup 9201 df-inf 9202 df-wsuc 33806 |
This theorem is referenced by: (None) |
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