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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 35794 | . 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) | |
2 | wsucex.1 | . . 3 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
3 | 2 | infexd 9521 | . 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 3478 Or wor 5596 ◡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 ax-un 7754 |
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-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 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-cnv 5697 df-sup 9480 df-inf 9481 df-wsuc 35794 |
This theorem is referenced by: (None) |
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