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| Mirrors > Home > MPE Home > Th. List > rex2dom | Structured version Visualization version GIF version | ||
| Description: A set that has at least 2 different members dominates ordinal 2. (Contributed by BTernaryTau, 30-Dec-2024.) |
| Ref | Expression |
|---|---|
| rex2dom | ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦) → 2o ≼ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elex 3501 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
| 2 | prssi 4821 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → {𝑥, 𝑦} ⊆ 𝐴) | |
| 3 | df2o3 8514 | . . . . . . . 8 ⊢ 2o = {∅, 1o} | |
| 4 | 0ex 5307 | . . . . . . . . . 10 ⊢ ∅ ∈ V | |
| 5 | 4 | a1i 11 | . . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 → ∅ ∈ V) |
| 6 | 1oex 8516 | . . . . . . . . . 10 ⊢ 1o ∈ V | |
| 7 | 6 | a1i 11 | . . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 → 1o ∈ V) |
| 8 | vex 3484 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
| 9 | 8 | a1i 11 | . . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 → 𝑥 ∈ V) |
| 10 | vex 3484 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
| 11 | 10 | a1i 11 | . . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 → 𝑦 ∈ V) |
| 12 | 1n0 8526 | . . . . . . . . . . 11 ⊢ 1o ≠ ∅ | |
| 13 | 12 | necomi 2995 | . . . . . . . . . 10 ⊢ ∅ ≠ 1o |
| 14 | 13 | a1i 11 | . . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 → ∅ ≠ 1o) |
| 15 | id 22 | . . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 → 𝑥 ≠ 𝑦) | |
| 16 | 5, 7, 9, 11, 14, 15 | en2prd 9088 | . . . . . . . 8 ⊢ (𝑥 ≠ 𝑦 → {∅, 1o} ≈ {𝑥, 𝑦}) |
| 17 | 3, 16 | eqbrtrid 5178 | . . . . . . 7 ⊢ (𝑥 ≠ 𝑦 → 2o ≈ {𝑥, 𝑦}) |
| 18 | endom 9019 | . . . . . . 7 ⊢ (2o ≈ {𝑥, 𝑦} → 2o ≼ {𝑥, 𝑦}) | |
| 19 | 17, 18 | syl 17 | . . . . . 6 ⊢ (𝑥 ≠ 𝑦 → 2o ≼ {𝑥, 𝑦}) |
| 20 | domssr 9039 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ {𝑥, 𝑦} ⊆ 𝐴 ∧ 2o ≼ {𝑥, 𝑦}) → 2o ≼ 𝐴) | |
| 21 | 20 | 3expib 1123 | . . . . . 6 ⊢ (𝐴 ∈ V → (({𝑥, 𝑦} ⊆ 𝐴 ∧ 2o ≼ {𝑥, 𝑦}) → 2o ≼ 𝐴)) |
| 22 | 2, 19, 21 | syl2ani 607 | . . . . 5 ⊢ (𝐴 ∈ V → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≠ 𝑦) → 2o ≼ 𝐴)) |
| 23 | 22 | expd 415 | . . . 4 ⊢ (𝐴 ∈ V → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ≠ 𝑦 → 2o ≼ 𝐴))) |
| 24 | 23 | rexlimdvv 3212 | . . 3 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 → 2o ≼ 𝐴)) |
| 25 | 1, 24 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 → 2o ≼ 𝐴)) |
| 26 | 25 | imp 406 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦) → 2o ≼ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 {cpr 4628 class class class wbr 5143 1oc1o 8499 2oc2o 8500 ≈ cen 8982 ≼ cdom 8983 |
| 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-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-suc 6390 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-1o 8506 df-2o 8507 df-en 8986 df-dom 8987 |
| This theorem is referenced by: 1sdom2dom 9283 1sdom 9284 |
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