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Theorem en0 8019

Description: The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.)

**Assertion**
Ref	Expression
en0	⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)

Proof of Theorem en0 Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 7964	. . 3 ⊢ (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴–1-1-onto→∅)
2		f1ocnv 6149	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→∅ → ^◡𝑓:∅–1-1-onto→𝐴)
3		f1o00 6171	. . . . . 6 ⊢ (^◡𝑓:∅–1-1-onto→𝐴 ↔ (^◡𝑓 = ∅ ∧ 𝐴 = ∅))
4	3	simprbi 480	. . . . 5 ⊢ (^◡𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅)
5	2, 4	syl 17	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅)
6	5	exlimiv 1858	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅)
7	1, 6	sylbi 207	. 2 ⊢ (𝐴 ≈ ∅ → 𝐴 = ∅)
8		0ex 4790	. . . 4 ⊢ ∅ ∈ V
9	8	enref 7988	. . 3 ⊢ ∅ ≈ ∅
10		breq1 4656	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅))
11	9, 10	mpbiri 248	. 2 ⊢ (𝐴 = ∅ → 𝐴 ≈ ∅)
12	7, 11	impbii 199	1 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 = wceq 1483 ∃wex 1704 ∅c0 3915 class class class wbr 4653 ^◡ccnv 5113 –1-1-onto→wf1o 5887 ≈ cen 7952

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-en 7956

This theorem is referenced by: snfi 8038 dom0 8088 0sdomg 8089 nneneq 8143 snnen2o 8149 enp1i 8195 findcard 8199 findcard2 8200 fiint 8237 cantnff 8571 cantnf0 8572 cantnfp1lem2 8576 cantnflem1 8586 cantnf 8590 cnfcom2lem 8598 cardnueq0 8790 infmap2 9040 fin23lem26 9147 cardeq0 9374 hasheq0 13154 mreexexd 16308 mreexexdOLD 16309 pmtrfmvdn0 17882 pmtrsn 17939 rp-isfinite6 37864