![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 6127 | . . . 4 ⊢ 2𝑜 = {∅, {∅}} | |
2 | 1 | breq1i 3818 | . . 3 ⊢ (2𝑜 ≼ 𝐴 ↔ {∅, {∅}} ≼ 𝐴) |
3 | brdomi 6395 | . . 3 ⊢ ({∅, {∅}} ≼ 𝐴 → ∃𝑓 𝑓:{∅, {∅}}–1-1→𝐴) | |
4 | 2, 3 | sylbi 119 | . 2 ⊢ (2𝑜 ≼ 𝐴 → ∃𝑓 𝑓:{∅, {∅}}–1-1→𝐴) |
5 | f1f 5163 | . . . . 5 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → 𝑓:{∅, {∅}}⟶𝐴) | |
6 | 0ex 3931 | . . . . . 6 ⊢ ∅ ∈ V | |
7 | 6 | prid1 3522 | . . . . 5 ⊢ ∅ ∈ {∅, {∅}} |
8 | ffvelrn 5376 | . . . . 5 ⊢ ((𝑓:{∅, {∅}}⟶𝐴 ∧ ∅ ∈ {∅, {∅}}) → (𝑓‘∅) ∈ 𝐴) | |
9 | 5, 7, 8 | sylancl 404 | . . . 4 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → (𝑓‘∅) ∈ 𝐴) |
10 | p0ex 3986 | . . . . . 6 ⊢ {∅} ∈ V | |
11 | 10 | prid2 3523 | . . . . 5 ⊢ {∅} ∈ {∅, {∅}} |
12 | ffvelrn 5376 | . . . . 5 ⊢ ((𝑓:{∅, {∅}}⟶𝐴 ∧ {∅} ∈ {∅, {∅}}) → (𝑓‘{∅}) ∈ 𝐴) | |
13 | 5, 11, 12 | sylancl 404 | . . . 4 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → (𝑓‘{∅}) ∈ 𝐴) |
14 | 0nep0 3965 | . . . . . 6 ⊢ ∅ ≠ {∅} | |
15 | 14 | neii 2251 | . . . . 5 ⊢ ¬ ∅ = {∅} |
16 | f1fveq 5490 | . . . . . 6 ⊢ ((𝑓:{∅, {∅}}–1-1→𝐴 ∧ (∅ ∈ {∅, {∅}} ∧ {∅} ∈ {∅, {∅}})) → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅})) | |
17 | 7, 11, 16 | mpanr12 430 | . . . . 5 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅})) |
18 | 15, 17 | mtbiri 633 | . . . 4 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → ¬ (𝑓‘∅) = (𝑓‘{∅})) |
19 | eqeq1 2089 | . . . . . 6 ⊢ (𝑥 = (𝑓‘∅) → (𝑥 = 𝑦 ↔ (𝑓‘∅) = 𝑦)) | |
20 | 19 | notbid 625 | . . . . 5 ⊢ (𝑥 = (𝑓‘∅) → (¬ 𝑥 = 𝑦 ↔ ¬ (𝑓‘∅) = 𝑦)) |
21 | eqeq2 2092 | . . . . . 6 ⊢ (𝑦 = (𝑓‘{∅}) → ((𝑓‘∅) = 𝑦 ↔ (𝑓‘∅) = (𝑓‘{∅}))) | |
22 | 21 | notbid 625 | . . . . 5 ⊢ (𝑦 = (𝑓‘{∅}) → (¬ (𝑓‘∅) = 𝑦 ↔ ¬ (𝑓‘∅) = (𝑓‘{∅}))) |
23 | 20, 22 | rspc2ev 2725 | . . . 4 ⊢ (((𝑓‘∅) ∈ 𝐴 ∧ (𝑓‘{∅}) ∈ 𝐴 ∧ ¬ (𝑓‘∅) = (𝑓‘{∅})) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦) |
24 | 9, 13, 18, 23 | syl3anc 1170 | . . 3 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦) |
25 | 24 | exlimiv 1530 | . 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 1285 ∃wex 1422 ∈ wcel 1434 ∃wrex 2354 ∅c0 3269 {csn 3422 {cpr 3423 class class class wbr 3811 ⟶wf 4964 –1-1→wf1 4965 ‘cfv 4968 2𝑜c2o 6106 ≼ cdom 6385 |
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 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 3999 ax-un 4223 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-v 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-id 4083 df-suc 4161 df-xp 4406 df-rel 4407 df-cnv 4408 df-co 4409 df-dm 4410 df-rn 4411 df-iota 4933 df-fun 4970 df-fn 4971 df-f 4972 df-f1 4973 df-fv 4976 df-1o 6112 df-2o 6113 df-dom 6388 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |