Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dtruALT2 | Structured version Visualization version GIF version |
Description: Alternate proof of dtru 4783 using ax-pr 4833 instead of ax-pow 4769. (Contributed by Mario Carneiro, 31-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dtruALT2 | ⊢ ¬ ∀𝑥 𝑥 = 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0inp0 4763 | . . . 4 ⊢ (𝑦 = ∅ → ¬ 𝑦 = {∅}) | |
2 | snex 4835 | . . . . 5 ⊢ {∅} ∈ V | |
3 | eqeq2 2621 | . . . . . 6 ⊢ (𝑥 = {∅} → (𝑦 = 𝑥 ↔ 𝑦 = {∅})) | |
4 | 3 | notbid 307 | . . . . 5 ⊢ (𝑥 = {∅} → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = {∅})) |
5 | 2, 4 | spcev 3273 | . . . 4 ⊢ (¬ 𝑦 = {∅} → ∃𝑥 ¬ 𝑦 = 𝑥) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥) |
7 | 0ex 4718 | . . . 4 ⊢ ∅ ∈ V | |
8 | eqeq2 2621 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑦 = 𝑥 ↔ 𝑦 = ∅)) | |
9 | 8 | notbid 307 | . . . 4 ⊢ (𝑥 = ∅ → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = ∅)) |
10 | 7, 9 | spcev 3273 | . . 3 ⊢ (¬ 𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥) |
11 | 6, 10 | pm2.61i 175 | . 2 ⊢ ∃𝑥 ¬ 𝑦 = 𝑥 |
12 | exnal 1744 | . . 3 ⊢ (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑦 = 𝑥) | |
13 | eqcom 2617 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
14 | 13 | albii 1737 | . . 3 ⊢ (∀𝑥 𝑦 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) |
15 | 12, 14 | xchbinx 323 | . 2 ⊢ (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑥 = 𝑦) |
16 | 11, 15 | mpbi 219 | 1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀wal 1473 = wceq 1475 ∃wex 1695 ∅c0 3874 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-un 3545 df-nul 3875 df-sn 4126 df-pr 4128 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |