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Mirrors > Home > MPE Home > Th. List > en1 | Structured version Visualization version GIF version |
Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.) |
Ref | Expression |
---|---|
en1 | ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df1o2 8116 | . . . . 5 ⊢ 1o = {∅} | |
2 | 1 | breq2i 5074 | . . . 4 ⊢ (𝐴 ≈ 1o ↔ 𝐴 ≈ {∅}) |
3 | bren 8518 | . . . 4 ⊢ (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴–1-1-onto→{∅}) | |
4 | 2, 3 | bitri 277 | . . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑓 𝑓:𝐴–1-1-onto→{∅}) |
5 | f1ocnv 6627 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→{∅} → ◡𝑓:{∅}–1-1-onto→𝐴) | |
6 | f1ofo 6622 | . . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}–onto→𝐴) | |
7 | forn 6593 | . . . . . . 7 ⊢ (◡𝑓:{∅}–onto→𝐴 → ran ◡𝑓 = 𝐴) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = 𝐴) |
9 | f1of 6615 | . . . . . . . . 9 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}⟶𝐴) | |
10 | 0ex 5211 | . . . . . . . . . . 11 ⊢ ∅ ∈ V | |
11 | 10 | fsn2 6898 | . . . . . . . . . 10 ⊢ (◡𝑓:{∅}⟶𝐴 ↔ ((◡𝑓‘∅) ∈ 𝐴 ∧ ◡𝑓 = {〈∅, (◡𝑓‘∅)〉})) |
12 | 11 | simprbi 499 | . . . . . . . . 9 ⊢ (◡𝑓:{∅}⟶𝐴 → ◡𝑓 = {〈∅, (◡𝑓‘∅)〉}) |
13 | 9, 12 | syl 17 | . . . . . . . 8 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓 = {〈∅, (◡𝑓‘∅)〉}) |
14 | 13 | rneqd 5808 | . . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = ran {〈∅, (◡𝑓‘∅)〉}) |
15 | 10 | rnsnop 6081 | . . . . . . 7 ⊢ ran {〈∅, (◡𝑓‘∅)〉} = {(◡𝑓‘∅)} |
16 | 14, 15 | syl6eq 2872 | . . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = {(◡𝑓‘∅)}) |
17 | 8, 16 | eqtr3d 2858 | . . . . 5 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → 𝐴 = {(◡𝑓‘∅)}) |
18 | fvex 6683 | . . . . . 6 ⊢ (◡𝑓‘∅) ∈ V | |
19 | sneq 4577 | . . . . . . 7 ⊢ (𝑥 = (◡𝑓‘∅) → {𝑥} = {(◡𝑓‘∅)}) | |
20 | 19 | eqeq2d 2832 | . . . . . 6 ⊢ (𝑥 = (◡𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(◡𝑓‘∅)})) |
21 | 18, 20 | spcev 3607 | . . . . 5 ⊢ (𝐴 = {(◡𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥}) |
22 | 5, 17, 21 | 3syl 18 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥}) |
23 | 22 | exlimiv 1931 | . . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥}) |
24 | 4, 23 | sylbi 219 | . 2 ⊢ (𝐴 ≈ 1o → ∃𝑥 𝐴 = {𝑥}) |
25 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
26 | 25 | ensn1 8573 | . . . 4 ⊢ {𝑥} ≈ 1o |
27 | breq1 5069 | . . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o)) | |
28 | 26, 27 | mpbiri 260 | . . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ≈ 1o) |
29 | 28 | exlimiv 1931 | . 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1o) |
30 | 24, 29 | impbii 211 | 1 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∅c0 4291 {csn 4567 〈cop 4573 class class class wbr 5066 ◡ccnv 5554 ran crn 5556 ⟶wf 6351 –onto→wfo 6353 –1-1-onto→wf1o 6354 ‘cfv 6355 1oc1o 8095 ≈ cen 8506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-1o 8102 df-en 8510 |
This theorem is referenced by: en1b 8577 reuen1 8578 en2 8754 card1 9397 pm54.43 9429 hash1elsn 13733 hash1snb 13781 ufildom1 22534 unidifsnel 30295 unidifsnne 30296 funen1cnv 32357 lfuhgr3 32366 snen1g 39939 |
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