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Mirrors > Home > ILE Home > Th. List > 1domsn | GIF version |
Description: A singleton (whether of a set or a proper class) is dominated by one. (Contributed by Jim Kingdon, 1-Mar-2022.) |
Ref | Expression |
---|---|
1domsn | ⊢ {𝐴} ≼ 1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0lt1o 6305 | . . . 4 ⊢ ∅ ∈ 1o | |
2 | 1 | rgenw 2464 | . . 3 ⊢ ∀𝑥 ∈ {𝐴}∅ ∈ 1o |
3 | elsni 3515 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
4 | 3 | adantr 274 | . . . . . 6 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝐴) |
5 | elsni 3515 | . . . . . . 7 ⊢ (𝑦 ∈ {𝐴} → 𝑦 = 𝐴) | |
6 | 5 | adantl 275 | . . . . . 6 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑦 = 𝐴) |
7 | 4, 6 | eqtr4d 2153 | . . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝑦) |
8 | 7 | a1d 22 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (∅ = ∅ → 𝑥 = 𝑦)) |
9 | 8 | rgen2a 2463 | . . 3 ⊢ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (∅ = ∅ → 𝑥 = 𝑦) |
10 | eqid 2117 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↦ ∅) = (𝑥 ∈ {𝐴} ↦ ∅) | |
11 | eqidd 2118 | . . . 4 ⊢ (𝑥 = 𝑦 → ∅ = ∅) | |
12 | 10, 11 | f1mpt 5640 | . . 3 ⊢ ((𝑥 ∈ {𝐴} ↦ ∅):{𝐴}–1-1→1o ↔ (∀𝑥 ∈ {𝐴}∅ ∈ 1o ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (∅ = ∅ → 𝑥 = 𝑦))) |
13 | 2, 9, 12 | mpbir2an 911 | . 2 ⊢ (𝑥 ∈ {𝐴} ↦ ∅):{𝐴}–1-1→1o |
14 | 1oex 6289 | . . 3 ⊢ 1o ∈ V | |
15 | 14 | f1dom 6622 | . 2 ⊢ ((𝑥 ∈ {𝐴} ↦ ∅):{𝐴}–1-1→1o → {𝐴} ≼ 1o) |
16 | 13, 15 | ax-mp 5 | 1 ⊢ {𝐴} ≼ 1o |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 ∀wral 2393 ∅c0 3333 {csn 3497 class class class wbr 3899 ↦ cmpt 3959 –1-1→wf1 5090 1oc1o 6274 ≼ cdom 6601 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-1o 6281 df-dom 6604 |
This theorem is referenced by: (None) |
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