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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 31756 | . 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) | |
2 | wsucex.1 | . . 3 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
3 | 2 | infexd 8389 | . 2 ⊢ (𝜑 → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ V) |
4 | 1, 3 | syl5eqel 2705 | 1 ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 Vcvv 3200 Or wor 5034 ◡ccnv 5113 Predcpred 5679 infcinf 8347 wsuccwsuc 31752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rmo 2920 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-po 5035 df-so 5036 df-cnv 5122 df-sup 8348 df-inf 8349 df-wsuc 31756 |
This theorem is referenced by: (None) |
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