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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 6964 | . . 3 ⊢ (𝐴 ∈ V → ( I ‘𝐴) = 𝐴) | |
2 | 1 | preq2d 4743 | . 2 ⊢ (𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴}) |
3 | dfsn2 4640 | . . . 4 ⊢ {∅} = {∅, ∅} | |
4 | 3 | eqcomi 2741 | . . 3 ⊢ {∅, ∅} = {∅} |
5 | fvprc 6880 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ( I ‘𝐴) = ∅) | |
6 | 5 | preq2d 4743 | . . 3 ⊢ (¬ 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, ∅}) |
7 | prprc2 4769 | . . 3 ⊢ (¬ 𝐴 ∈ V → {∅, 𝐴} = {∅}) | |
8 | 4, 6, 7 | 3eqtr4a 2798 | . 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 1541 ∈ wcel 2106 Vcvv 3474 ∅c0 4321 {csn 4627 {cpr 4629 I cid 5572 ‘cfv 6540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 |
This theorem is referenced by: indistop 22496 indisuni 22497 indiscld 22586 indisconn 22913 txindis 23129 hmphindis 23292 |
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