![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 6242 | . . . 4 ⊢ ∅ ∈ 1o | |
2 | 1 | rgenw 2441 | . . 3 ⊢ ∀𝑥 ∈ {𝐴}∅ ∈ 1o |
3 | elsni 3484 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
4 | 3 | adantr 271 | . . . . . 6 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝐴) |
5 | elsni 3484 | . . . . . . 7 ⊢ (𝑦 ∈ {𝐴} → 𝑦 = 𝐴) | |
6 | 5 | adantl 272 | . . . . . 6 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑦 = 𝐴) |
7 | 4, 6 | eqtr4d 2130 | . . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝑦) |
8 | 7 | a1d 22 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (∅ = ∅ → 𝑥 = 𝑦)) |
9 | 8 | rgen2a 2440 | . . 3 ⊢ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (∅ = ∅ → 𝑥 = 𝑦) |
10 | eqid 2095 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↦ ∅) = (𝑥 ∈ {𝐴} ↦ ∅) | |
11 | eqidd 2096 | . . . 4 ⊢ (𝑥 = 𝑦 → ∅ = ∅) | |
12 | 10, 11 | f1mpt 5588 | . . 3 ⊢ ((𝑥 ∈ {𝐴} ↦ ∅):{𝐴}–1-1→1o ↔ (∀𝑥 ∈ {𝐴}∅ ∈ 1o ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (∅ = ∅ → 𝑥 = 𝑦))) |
13 | 2, 9, 12 | mpbir2an 891 | . 2 ⊢ (𝑥 ∈ {𝐴} ↦ ∅):{𝐴}–1-1→1o |
14 | 1oex 6227 | . . 3 ⊢ 1o ∈ V | |
15 | 14 | f1dom 6557 | . 2 ⊢ ((𝑥 ∈ {𝐴} ↦ ∅):{𝐴}–1-1→1o → {𝐴} ≼ 1o) |
16 | 13, 15 | ax-mp 7 | 1 ⊢ {𝐴} ≼ 1o |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1296 ∈ wcel 1445 ∀wral 2370 ∅c0 3302 {csn 3466 class class class wbr 3867 ↦ cmpt 3921 –1-1→wf1 5046 1oc1o 6212 ≼ cdom 6536 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-id 4144 df-iord 4217 df-on 4219 df-suc 4222 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-1o 6219 df-dom 6539 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |