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Mirrors > Home > MPE Home > Th. List > indislem | Structured version Visualization version GIF version |
Description: A lemma to eliminate some sethood hypotheses when dealing with the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
indislem | ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvi 6502 | . . 3 ⊢ (𝐴 ∈ V → ( I ‘𝐴) = 𝐴) | |
2 | 1 | preq2d 4493 | . 2 ⊢ (𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴}) |
3 | dfsn2 4410 | . . . 4 ⊢ {∅} = {∅, ∅} | |
4 | 3 | eqcomi 2834 | . . 3 ⊢ {∅, ∅} = {∅} |
5 | fvprc 6426 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ( I ‘𝐴) = ∅) | |
6 | 5 | preq2d 4493 | . . 3 ⊢ (¬ 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, ∅}) |
7 | prprc2 4519 | . . 3 ⊢ (¬ 𝐴 ∈ V → {∅, 𝐴} = {∅}) | |
8 | 4, 6, 7 | 3eqtr4a 2887 | . 2 ⊢ (¬ 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴}) |
9 | 2, 8 | pm2.61i 177 | 1 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ∅c0 4144 {csn 4397 {cpr 4399 I cid 5249 ‘cfv 6123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-iota 6086 df-fun 6125 df-fv 6131 |
This theorem is referenced by: indistop 21177 indisuni 21178 indiscld 21266 indisconn 21592 txindis 21808 hmphindis 21971 |
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