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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 6961 | . . 3 ⊢ (𝐴 ∈ V → ( I ‘𝐴) = 𝐴) | |
2 | 1 | preq2d 4739 | . 2 ⊢ (𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴}) |
3 | dfsn2 4636 | . . . 4 ⊢ {∅} = {∅, ∅} | |
4 | 3 | eqcomi 2735 | . . 3 ⊢ {∅, ∅} = {∅} |
5 | fvprc 6877 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ( I ‘𝐴) = ∅) | |
6 | 5 | preq2d 4739 | . . 3 ⊢ (¬ 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, ∅}) |
7 | prprc2 4765 | . . 3 ⊢ (¬ 𝐴 ∈ V → {∅, 𝐴} = {∅}) | |
8 | 4, 6, 7 | 3eqtr4a 2792 | . 2 ⊢ (¬ 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴}) |
9 | 2, 8 | pm2.61i 182 | 1 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ∅c0 4317 {csn 4623 {cpr 4625 I cid 5566 ‘cfv 6537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6489 df-fun 6539 df-fv 6545 |
This theorem is referenced by: indistop 22860 indisuni 22861 indiscld 22950 indisconn 23277 txindis 23493 hmphindis 23656 |
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