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Mirrors > Home > MPE Home > Th. List > omsucelsucb | Structured version Visualization version GIF version |
Description: Membership is inherited by successors for natural numbers. (Contributed by AV, 15-Sep-2023.) |
Ref | Expression |
---|---|
omsucelsucb | ⊢ (𝑁 ∈ ω ↔ suc 𝑁 ∈ suc ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordom 7803 | . 2 ⊢ Ord ω | |
2 | ordsucelsuc 7748 | . 2 ⊢ (Ord ω → (𝑁 ∈ ω ↔ suc 𝑁 ∈ suc ω)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝑁 ∈ ω ↔ suc 𝑁 ∈ suc ω) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2107 Ord word 6313 suc csuc 6316 ωcom 7793 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-un 7663 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-tr 5222 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-om 7794 |
This theorem is referenced by: satf0suc 33750 sat1el2xp 33753 fmlasuc0 33758 |
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