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Mirrors > Home > MPE Home > Th. List > dtruALT2 | Structured version Visualization version GIF version |
Description: Alternate proof of dtru 5238 using ax-pr 5297 instead of ax-pow 5233. (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 5226 | . . . 4 ⊢ (𝑦 = ∅ → ¬ 𝑦 = {∅}) | |
2 | snex 5299 | . . . . 5 ⊢ {∅} ∈ V | |
3 | eqeq2 2751 | . . . . . 6 ⊢ (𝑥 = {∅} → (𝑦 = 𝑥 ↔ 𝑦 = {∅})) | |
4 | 3 | notbid 321 | . . . . 5 ⊢ (𝑥 = {∅} → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = {∅})) |
5 | 2, 4 | spcev 3511 | . . . 4 ⊢ (¬ 𝑦 = {∅} → ∃𝑥 ¬ 𝑦 = 𝑥) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥) |
7 | 0ex 5176 | . . . 4 ⊢ ∅ ∈ V | |
8 | eqeq2 2751 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑦 = 𝑥 ↔ 𝑦 = ∅)) | |
9 | 8 | notbid 321 | . . . 4 ⊢ (𝑥 = ∅ → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = ∅)) |
10 | 7, 9 | spcev 3511 | . . 3 ⊢ (¬ 𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥) |
11 | 6, 10 | pm2.61i 185 | . 2 ⊢ ∃𝑥 ¬ 𝑦 = 𝑥 |
12 | exnal 1833 | . . 3 ⊢ (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑦 = 𝑥) | |
13 | eqcom 2746 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
14 | 13 | albii 1826 | . . 3 ⊢ (∀𝑥 𝑦 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) |
15 | 12, 14 | xchbinx 337 | . 2 ⊢ (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑥 = 𝑦) |
16 | 11, 15 | mpbi 233 | 1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀wal 1540 = wceq 1542 ∃wex 1786 ∅c0 4212 {csn 4517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pr 5297 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-ne 2936 df-v 3401 df-dif 3847 df-un 3849 df-nul 4213 df-sn 4518 df-pr 4520 |
This theorem is referenced by: fvprc 6669 |
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