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Mirrors > Home > MPE Home > Th. List > Mathboxes > epelon2 | Structured version Visualization version GIF version |
Description: Over the ordinal numbers, one may define the relation 𝐴 E 𝐵 iff 𝐴 ∈ 𝐵 and one finds that, under this ordering, On is a well-ordered class, see epweon 7494. This is a weak form of epelg 5463 which only requires that we know 𝐵 to be a set. (Contributed by RP, 27-Sep-2023.) |
Ref | Expression |
---|---|
epelon2 | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | epelg 5463 | . 2 ⊢ (𝐵 ∈ On → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
2 | 1 | adantl 484 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2113 class class class wbr 5063 E cep 5461 Oncon0 6188 |
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 5200 ax-nul 5207 ax-pr 5327 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-rab 3146 df-v 3495 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-sn 4565 df-pr 4567 df-op 4571 df-br 5064 df-opab 5126 df-eprel 5462 |
This theorem is referenced by: (None) |
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