Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7582 | . 2 ⊢ Ord ω | |
2 | ordsucelsuc 7530 | . 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 208 ∈ wcel 2113 Ord word 6183 suc csuc 6186 ωcom 7573 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-tr 5166 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-om 7574 |
This theorem is referenced by: satf0suc 32644 sat1el2xp 32647 fmlasuc0 32652 |
Copyright terms: Public domain | W3C validator |