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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 3498 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | prssi 4825 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → {𝑥, 𝑦} ⊆ 𝐴) | |
3 | df2o3 8512 | . . . . . . . 8 ⊢ 2o = {∅, 1o} | |
4 | 0ex 5312 | . . . . . . . . . 10 ⊢ ∅ ∈ V | |
5 | 4 | a1i 11 | . . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 → ∅ ∈ V) |
6 | 1oex 8514 | . . . . . . . . . 10 ⊢ 1o ∈ V | |
7 | 6 | a1i 11 | . . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 → 1o ∈ V) |
8 | vex 3481 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
9 | 8 | a1i 11 | . . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 → 𝑥 ∈ V) |
10 | vex 3481 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
11 | 10 | a1i 11 | . . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 → 𝑦 ∈ V) |
12 | 1n0 8524 | . . . . . . . . . . 11 ⊢ 1o ≠ ∅ | |
13 | 12 | necomi 2992 | . . . . . . . . . 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 9086 | . . . . . . . 8 ⊢ (𝑥 ≠ 𝑦 → {∅, 1o} ≈ {𝑥, 𝑦}) |
17 | 3, 16 | eqbrtrid 5182 | . . . . . . 7 ⊢ (𝑥 ≠ 𝑦 → 2o ≈ {𝑥, 𝑦}) |
18 | endom 9017 | . . . . . . 7 ⊢ (2o ≈ {𝑥, 𝑦} → 2o ≼ {𝑥, 𝑦}) | |
19 | 17, 18 | syl 17 | . . . . . 6 ⊢ (𝑥 ≠ 𝑦 → 2o ≼ {𝑥, 𝑦}) |
20 | domssr 9037 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ {𝑥, 𝑦} ⊆ 𝐴 ∧ 2o ≼ {𝑥, 𝑦}) → 2o ≼ 𝐴) | |
21 | 20 | 3expib 1121 | . . . . . 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 3209 | . . 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 2105 ≠ wne 2937 ∃wrex 3067 Vcvv 3477 ⊆ wss 3962 ∅c0 4338 {cpr 4632 class class class wbr 5147 1oc1o 8497 2oc2o 8498 ≈ cen 8980 ≼ cdom 8981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-suc 6391 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-1o 8504 df-2o 8505 df-en 8984 df-dom 8985 |
This theorem is referenced by: 1sdom2dom 9280 1sdom 9281 |
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