Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6734 | . . 3 ⊢ (𝐴 ∈ V → ( I ‘𝐴) = 𝐴) | |
2 | 1 | preq2d 4669 | . 2 ⊢ (𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴}) |
3 | dfsn2 4573 | . . . 4 ⊢ {∅} = {∅, ∅} | |
4 | 3 | eqcomi 2830 | . . 3 ⊢ {∅, ∅} = {∅} |
5 | fvprc 6657 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ( I ‘𝐴) = ∅) | |
6 | 5 | preq2d 4669 | . . 3 ⊢ (¬ 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, ∅}) |
7 | prprc2 4695 | . . 3 ⊢ (¬ 𝐴 ∈ V → {∅, 𝐴} = {∅}) | |
8 | 4, 6, 7 | 3eqtr4a 2882 | . 2 ⊢ (¬ 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴}) |
9 | 2, 8 | pm2.61i 184 | 1 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 {csn 4560 {cpr 4562 I cid 5453 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 |
This theorem is referenced by: indistop 21604 indisuni 21605 indiscld 21693 indisconn 22020 txindis 22236 hmphindis 22399 |
Copyright terms: Public domain | W3C validator |