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Mirrors > Home > MPE Home > Th. List > 1sdom | Structured version Visualization version GIF version |
Description: A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 8981.) (Contributed by Mario Carneiro, 12-Jan-2013.) Avoid ax-un 7677. (Revised by BTernaryTau, 30-Dec-2024.) |
Ref | Expression |
---|---|
1sdom | ⊢ (𝐴 ∈ 𝑉 → (1o ≺ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1sdom2dom 9198 | . 2 ⊢ (1o ≺ 𝐴 ↔ 2o ≼ 𝐴) | |
2 | 2dom 8981 | . . 3 ⊢ (2o ≼ 𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦) | |
3 | df-ne 2945 | . . . . . 6 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦) | |
4 | 3 | 2rexbii 3129 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦) |
5 | rex2dom 9197 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦) → 2o ≼ 𝐴) | |
6 | 4, 5 | sylan2br 596 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦) → 2o ≼ 𝐴) |
7 | 6 | ex 414 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦 → 2o ≼ 𝐴)) |
8 | 2, 7 | impbid2 225 | . 2 ⊢ (𝐴 ∈ 𝑉 → (2o ≼ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦)) |
9 | 1, 8 | bitrid 283 | 1 ⊢ (𝐴 ∈ 𝑉 → (1o ≺ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∈ wcel 2107 ≠ wne 2944 ∃wrex 3074 class class class wbr 5110 1oc1o 8410 2oc2o 8411 ≼ cdom 8888 ≺ csdm 8889 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-1o 8417 df-2o 8418 df-en 8891 df-dom 8892 df-sdom 8893 |
This theorem is referenced by: unxpdomlem3 9203 frgpnabl 19660 isnzr2 20749 |
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