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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 3487 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | prssi 4819 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → {𝑥, 𝑦} ⊆ 𝐴) | |
3 | df2o3 8472 | . . . . . . . 8 ⊢ 2o = {∅, 1o} | |
4 | 0ex 5300 | . . . . . . . . . 10 ⊢ ∅ ∈ V | |
5 | 4 | a1i 11 | . . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 → ∅ ∈ V) |
6 | 1oex 8474 | . . . . . . . . . 10 ⊢ 1o ∈ V | |
7 | 6 | a1i 11 | . . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 → 1o ∈ V) |
8 | vex 3472 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
9 | 8 | a1i 11 | . . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 → 𝑥 ∈ V) |
10 | vex 3472 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
11 | 10 | a1i 11 | . . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 → 𝑦 ∈ V) |
12 | 1n0 8486 | . . . . . . . . . . 11 ⊢ 1o ≠ ∅ | |
13 | 12 | necomi 2989 | . . . . . . . . . 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 9047 | . . . . . . . 8 ⊢ (𝑥 ≠ 𝑦 → {∅, 1o} ≈ {𝑥, 𝑦}) |
17 | 3, 16 | eqbrtrid 5176 | . . . . . . 7 ⊢ (𝑥 ≠ 𝑦 → 2o ≈ {𝑥, 𝑦}) |
18 | endom 8974 | . . . . . . 7 ⊢ (2o ≈ {𝑥, 𝑦} → 2o ≼ {𝑥, 𝑦}) | |
19 | 17, 18 | syl 17 | . . . . . 6 ⊢ (𝑥 ≠ 𝑦 → 2o ≼ {𝑥, 𝑦}) |
20 | domssr 8994 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ {𝑥, 𝑦} ⊆ 𝐴 ∧ 2o ≼ {𝑥, 𝑦}) → 2o ≼ 𝐴) | |
21 | 20 | 3expib 1119 | . . . . . 6 ⊢ (𝐴 ∈ V → (({𝑥, 𝑦} ⊆ 𝐴 ∧ 2o ≼ {𝑥, 𝑦}) → 2o ≼ 𝐴)) |
22 | 2, 19, 21 | syl2ani 606 | . . . . 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 3204 | . . 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 2098 ≠ wne 2934 ∃wrex 3064 Vcvv 3468 ⊆ wss 3943 ∅c0 4317 {cpr 4625 class class class wbr 5141 1oc1o 8457 2oc2o 8458 ≈ cen 8935 ≼ cdom 8936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-suc 6363 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-1o 8464 df-2o 8465 df-en 8939 df-dom 8940 |
This theorem is referenced by: 1sdom2dom 9246 1sdom 9247 |
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