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| Mirrors > Home > MPE Home > Th. List > Mathboxes > omge2 | Structured version Visualization version GIF version | ||
| Description: Any non-zero ordinal product is greater-than-or-equal to the term on the right. Lemma 3.12 of [Schloeder] p. 9. See omword2 8612. (Contributed by RP, 29-Jan-2025.) |
| Ref | Expression |
|---|---|
| omge2 | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐵 ⊆ (𝐴 ·o 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ancom 460 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ↔ (𝐵 ∈ On ∧ 𝐴 ∈ On)) | |
| 2 | 1 | anbi1i 624 | . . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ≠ ∅) ↔ ((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐴 ≠ ∅)) |
| 3 | df-3an 1089 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ↔ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ≠ ∅)) | |
| 4 | on0eln0 6440 | . . . . 5 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
| 5 | 4 | adantl 481 | . . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
| 6 | 5 | pm5.32i 574 | . . 3 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) ↔ ((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐴 ≠ ∅)) |
| 7 | 2, 3, 6 | 3bitr4i 303 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ↔ ((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴)) |
| 8 | omword2 8612 | . 2 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → 𝐵 ⊆ (𝐴 ·o 𝐵)) | |
| 9 | 7, 8 | sylbi 217 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐵 ⊆ (𝐴 ·o 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 ∅c0 4333 Oncon0 6384 (class class class)co 7431 ·o comu 8504 |
| 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-rep 5279 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-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-omul 8511 |
| This theorem is referenced by: (None) |
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