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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 6918 | . . 3 ⊢ (𝐴 ∈ V → ( I ‘𝐴) = 𝐴) | |
2 | 1 | preq2d 4702 | . 2 ⊢ (𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴}) |
3 | dfsn2 4600 | . . . 4 ⊢ {∅} = {∅, ∅} | |
4 | 3 | eqcomi 2742 | . . 3 ⊢ {∅, ∅} = {∅} |
5 | fvprc 6835 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ( I ‘𝐴) = ∅) | |
6 | 5 | preq2d 4702 | . . 3 ⊢ (¬ 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, ∅}) |
7 | prprc2 4728 | . . 3 ⊢ (¬ 𝐴 ∈ V → {∅, 𝐴} = {∅}) | |
8 | 4, 6, 7 | 3eqtr4a 2799 | . 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 1542 ∈ wcel 2107 Vcvv 3444 ∅c0 4283 {csn 4587 {cpr 4589 I cid 5531 ‘cfv 6497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 |
This theorem is referenced by: indistop 22368 indisuni 22369 indiscld 22458 indisconn 22785 txindis 23001 hmphindis 23164 |
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