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Mirrors > Home > ILE Home > Th. List > 2dom | GIF version |
Description: A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.) |
Ref | Expression |
---|---|
2dom | ⊢ (2𝑜 ≼ 𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df2o2 6188 | . . . 4 ⊢ 2𝑜 = {∅, {∅}} | |
2 | 1 | breq1i 3850 | . . 3 ⊢ (2𝑜 ≼ 𝐴 ↔ {∅, {∅}} ≼ 𝐴) |
3 | brdomi 6456 | . . 3 ⊢ ({∅, {∅}} ≼ 𝐴 → ∃𝑓 𝑓:{∅, {∅}}–1-1→𝐴) | |
4 | 2, 3 | sylbi 119 | . 2 ⊢ (2𝑜 ≼ 𝐴 → ∃𝑓 𝑓:{∅, {∅}}–1-1→𝐴) |
5 | f1f 5210 | . . . . 5 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → 𝑓:{∅, {∅}}⟶𝐴) | |
6 | 0ex 3964 | . . . . . 6 ⊢ ∅ ∈ V | |
7 | 6 | prid1 3546 | . . . . 5 ⊢ ∅ ∈ {∅, {∅}} |
8 | ffvelrn 5426 | . . . . 5 ⊢ ((𝑓:{∅, {∅}}⟶𝐴 ∧ ∅ ∈ {∅, {∅}}) → (𝑓‘∅) ∈ 𝐴) | |
9 | 5, 7, 8 | sylancl 404 | . . . 4 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → (𝑓‘∅) ∈ 𝐴) |
10 | p0ex 4021 | . . . . . 6 ⊢ {∅} ∈ V | |
11 | 10 | prid2 3547 | . . . . 5 ⊢ {∅} ∈ {∅, {∅}} |
12 | ffvelrn 5426 | . . . . 5 ⊢ ((𝑓:{∅, {∅}}⟶𝐴 ∧ {∅} ∈ {∅, {∅}}) → (𝑓‘{∅}) ∈ 𝐴) | |
13 | 5, 11, 12 | sylancl 404 | . . . 4 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → (𝑓‘{∅}) ∈ 𝐴) |
14 | 0nep0 3998 | . . . . . 6 ⊢ ∅ ≠ {∅} | |
15 | 14 | neii 2257 | . . . . 5 ⊢ ¬ ∅ = {∅} |
16 | f1fveq 5543 | . . . . . 6 ⊢ ((𝑓:{∅, {∅}}–1-1→𝐴 ∧ (∅ ∈ {∅, {∅}} ∧ {∅} ∈ {∅, {∅}})) → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅})) | |
17 | 7, 11, 16 | mpanr12 430 | . . . . 5 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅})) |
18 | 15, 17 | mtbiri 635 | . . . 4 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → ¬ (𝑓‘∅) = (𝑓‘{∅})) |
19 | eqeq1 2094 | . . . . . 6 ⊢ (𝑥 = (𝑓‘∅) → (𝑥 = 𝑦 ↔ (𝑓‘∅) = 𝑦)) | |
20 | 19 | notbid 627 | . . . . 5 ⊢ (𝑥 = (𝑓‘∅) → (¬ 𝑥 = 𝑦 ↔ ¬ (𝑓‘∅) = 𝑦)) |
21 | eqeq2 2097 | . . . . . 6 ⊢ (𝑦 = (𝑓‘{∅}) → ((𝑓‘∅) = 𝑦 ↔ (𝑓‘∅) = (𝑓‘{∅}))) | |
22 | 21 | notbid 627 | . . . . 5 ⊢ (𝑦 = (𝑓‘{∅}) → (¬ (𝑓‘∅) = 𝑦 ↔ ¬ (𝑓‘∅) = (𝑓‘{∅}))) |
23 | 20, 22 | rspc2ev 2736 | . . . 4 ⊢ (((𝑓‘∅) ∈ 𝐴 ∧ (𝑓‘{∅}) ∈ 𝐴 ∧ ¬ (𝑓‘∅) = (𝑓‘{∅})) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦) |
24 | 9, 13, 18, 23 | syl3anc 1174 | . . 3 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦) |
25 | 24 | exlimiv 1534 | . 2 ⊢ (∃𝑓 𝑓:{∅, {∅}}–1-1→𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦) |
26 | 4, 25 | syl 14 | 1 ⊢ (2𝑜 ≼ 𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 103 = wceq 1289 ∃wex 1426 ∈ wcel 1438 ∃wrex 2360 ∅c0 3286 {csn 3444 {cpr 3445 class class class wbr 3843 ⟶wf 5006 –1-1→wf1 5007 ‘cfv 5010 2𝑜c2o 6167 ≼ cdom 6446 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3955 ax-nul 3963 ax-pow 4007 ax-pr 4034 ax-un 4258 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3429 df-sn 3450 df-pr 3451 df-op 3453 df-uni 3652 df-br 3844 df-opab 3898 df-id 4118 df-suc 4196 df-xp 4442 df-rel 4443 df-cnv 4444 df-co 4445 df-dm 4446 df-rn 4447 df-iota 4975 df-fun 5012 df-fn 5013 df-f 5014 df-f1 5015 df-fv 5018 df-1o 6173 df-2o 6174 df-dom 6449 |
This theorem is referenced by: (None) |
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