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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 35813 | . 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) | |
| 2 | wsucex.1 | . . 3 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
| 3 | 2 | infexd 9523 | . 2 ⊢ (𝜑 → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ V) |
| 4 | 1, 3 | eqeltrid 2845 | 1 ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 Or wor 5591 ◡ccnv 5684 Predcpred 6320 infcinf 9481 wsuccwsuc 35811 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-po 5592 df-so 5593 df-cnv 5693 df-sup 9482 df-inf 9483 df-wsuc 35813 |
| This theorem is referenced by: (None) |
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