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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 6345 | . . . 4 ⊢ ∅ ∈ 1o | |
2 | 1 | rgenw 2490 | . . 3 ⊢ ∀𝑥 ∈ {𝐴}∅ ∈ 1o |
3 | elsni 3550 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
4 | 3 | adantr 274 | . . . . . 6 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝐴) |
5 | elsni 3550 | . . . . . . 7 ⊢ (𝑦 ∈ {𝐴} → 𝑦 = 𝐴) | |
6 | 5 | adantl 275 | . . . . . 6 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑦 = 𝐴) |
7 | 4, 6 | eqtr4d 2176 | . . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝑦) |
8 | 7 | a1d 22 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (∅ = ∅ → 𝑥 = 𝑦)) |
9 | 8 | rgen2a 2489 | . . 3 ⊢ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (∅ = ∅ → 𝑥 = 𝑦) |
10 | eqid 2140 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↦ ∅) = (𝑥 ∈ {𝐴} ↦ ∅) | |
11 | eqidd 2141 | . . . 4 ⊢ (𝑥 = 𝑦 → ∅ = ∅) | |
12 | 10, 11 | f1mpt 5680 | . . 3 ⊢ ((𝑥 ∈ {𝐴} ↦ ∅):{𝐴}–1-1→1o ↔ (∀𝑥 ∈ {𝐴}∅ ∈ 1o ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (∅ = ∅ → 𝑥 = 𝑦))) |
13 | 2, 9, 12 | mpbir2an 927 | . 2 ⊢ (𝑥 ∈ {𝐴} ↦ ∅):{𝐴}–1-1→1o |
14 | 1oex 6329 | . . 3 ⊢ 1o ∈ V | |
15 | 14 | f1dom 6662 | . 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 1332 ∈ wcel 1481 ∀wral 2417 ∅c0 3368 {csn 3532 class class class wbr 3937 ↦ cmpt 3997 –1-1→wf1 5128 1oc1o 6314 ≼ cdom 6641 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-1o 6321 df-dom 6644 |
This theorem is referenced by: (None) |
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