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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 6328 | . . . 4 ⊢ 2o = {∅, {∅}} | |
2 | 1 | breq1i 3936 | . . 3 ⊢ (2o ≼ 𝐴 ↔ {∅, {∅}} ≼ 𝐴) |
3 | brdomi 6643 | . . 3 ⊢ ({∅, {∅}} ≼ 𝐴 → ∃𝑓 𝑓:{∅, {∅}}–1-1→𝐴) | |
4 | 2, 3 | sylbi 120 | . 2 ⊢ (2o ≼ 𝐴 → ∃𝑓 𝑓:{∅, {∅}}–1-1→𝐴) |
5 | f1f 5328 | . . . . 5 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → 𝑓:{∅, {∅}}⟶𝐴) | |
6 | 0ex 4055 | . . . . . 6 ⊢ ∅ ∈ V | |
7 | 6 | prid1 3629 | . . . . 5 ⊢ ∅ ∈ {∅, {∅}} |
8 | ffvelrn 5553 | . . . . 5 ⊢ ((𝑓:{∅, {∅}}⟶𝐴 ∧ ∅ ∈ {∅, {∅}}) → (𝑓‘∅) ∈ 𝐴) | |
9 | 5, 7, 8 | sylancl 409 | . . . 4 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → (𝑓‘∅) ∈ 𝐴) |
10 | p0ex 4112 | . . . . . 6 ⊢ {∅} ∈ V | |
11 | 10 | prid2 3630 | . . . . 5 ⊢ {∅} ∈ {∅, {∅}} |
12 | ffvelrn 5553 | . . . . 5 ⊢ ((𝑓:{∅, {∅}}⟶𝐴 ∧ {∅} ∈ {∅, {∅}}) → (𝑓‘{∅}) ∈ 𝐴) | |
13 | 5, 11, 12 | sylancl 409 | . . . 4 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → (𝑓‘{∅}) ∈ 𝐴) |
14 | 0nep0 4089 | . . . . . 6 ⊢ ∅ ≠ {∅} | |
15 | 14 | neii 2310 | . . . . 5 ⊢ ¬ ∅ = {∅} |
16 | f1fveq 5673 | . . . . . 6 ⊢ ((𝑓:{∅, {∅}}–1-1→𝐴 ∧ (∅ ∈ {∅, {∅}} ∧ {∅} ∈ {∅, {∅}})) → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅})) | |
17 | 7, 11, 16 | mpanr12 435 | . . . . 5 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅})) |
18 | 15, 17 | mtbiri 664 | . . . 4 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → ¬ (𝑓‘∅) = (𝑓‘{∅})) |
19 | eqeq1 2146 | . . . . . 6 ⊢ (𝑥 = (𝑓‘∅) → (𝑥 = 𝑦 ↔ (𝑓‘∅) = 𝑦)) | |
20 | 19 | notbid 656 | . . . . 5 ⊢ (𝑥 = (𝑓‘∅) → (¬ 𝑥 = 𝑦 ↔ ¬ (𝑓‘∅) = 𝑦)) |
21 | eqeq2 2149 | . . . . . 6 ⊢ (𝑦 = (𝑓‘{∅}) → ((𝑓‘∅) = 𝑦 ↔ (𝑓‘∅) = (𝑓‘{∅}))) | |
22 | 21 | notbid 656 | . . . . 5 ⊢ (𝑦 = (𝑓‘{∅}) → (¬ (𝑓‘∅) = 𝑦 ↔ ¬ (𝑓‘∅) = (𝑓‘{∅}))) |
23 | 20, 22 | rspc2ev 2804 | . . . 4 ⊢ (((𝑓‘∅) ∈ 𝐴 ∧ (𝑓‘{∅}) ∈ 𝐴 ∧ ¬ (𝑓‘∅) = (𝑓‘{∅})) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦) |
24 | 9, 13, 18, 23 | syl3anc 1216 | . . 3 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦) |
25 | 24 | exlimiv 1577 | . 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 104 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ∃wrex 2417 ∅c0 3363 {csn 3527 {cpr 3528 class class class wbr 3929 ⟶wf 5119 –1-1→wf1 5120 ‘cfv 5123 2oc2o 6307 ≼ cdom 6633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fv 5131 df-1o 6313 df-2o 6314 df-dom 6636 |
This theorem is referenced by: (None) |
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