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Mirrors > Home > MPE Home > Th. List > enref | Structured version Visualization version GIF version |
Description: Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.) |
Ref | Expression |
---|---|
enref.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
enref | ⊢ 𝐴 ≈ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enref.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | enrefg 7987 | . 2 ⊢ (𝐴 ∈ V → 𝐴 ≈ 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐴 ≈ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 Vcvv 3200 class class class wbr 4653 ≈ 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: ener 8002 enerOLD 8003 en0 8019 pwen 8133 phplem2 8140 phplem3 8141 isinf 8173 pssnn 8178 karden 8758 mappwen 8935 cdacomen 9003 infmap2 9040 ackbij1lem5 9046 axcc4dom 9263 domtriomlem 9264 cfpwsdom 9406 0tsk 9577 fzennn 12767 qnnen 14942 rpnnen 14956 rexpen 14957 lmisfree 20181 met2ndci 22327 lgseisenlem2 25101 poimirlem9 33418 poimirlem26 33435 |
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