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Mirrors > Home > MPE Home > Th. List > Mathboxes > omge1 | Structured version Visualization version GIF version |
Description: Any non-zero ordinal product is greater-than-or-equal to the term on the left. Lemma 3.11 of [Schloeder] p. 8. See omword1 8558. (Contributed by RP, 29-Jan-2025.) |
Ref | Expression |
---|---|
omge1 | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ≠ ∅) → 𝐴 ⊆ (𝐴 ·o 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpa 1148 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ≠ ∅) → (𝐴 ∈ On ∧ 𝐵 ∈ On)) | |
2 | on0eln0 6410 | . . . 4 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅)) | |
3 | 2 | biimpar 478 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅) → ∅ ∈ 𝐵) |
4 | 3 | 3adant1 1130 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ≠ ∅) → ∅ ∈ 𝐵) |
5 | omword1 8558 | . 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·o 𝐵)) | |
6 | 1, 4, 5 | syl2anc 584 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ≠ ∅) → 𝐴 ⊆ (𝐴 ·o 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 ∈ wcel 2106 ≠ wne 2940 ⊆ wss 3945 ∅c0 4319 Oncon0 6354 (class class class)co 7394 ·o comu 8448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pr 5421 ax-un 7709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7840 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-1o 8450 df-oadd 8454 df-omul 8455 |
This theorem is referenced by: (None) |
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