Theorem dtruALT2 5301

Description: Alternate proof of dtru 5236 using ax-pr 5295 instead of ax-pow 5231. (Contributed by Mario Carneiro, 31-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dtruALT2	⊢ ¬ ∀𝑥 𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem dtruALT2

Step	Hyp	Ref	Expression
1		0inp0 5224	. . . 4 ⊢ (𝑦 = ∅ → ¬ 𝑦 = {∅})
2		snex 5297	. . . . 5 ⊢ {∅} ∈ V
3		eqeq2 2810	. . . . . 6 ⊢ (𝑥 = {∅} → (𝑦 = 𝑥 ↔ 𝑦 = {∅}))
4	3	notbid 321	. . . . 5 ⊢ (𝑥 = {∅} → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = {∅}))
5	2, 4	spcev 3555	. . . 4 ⊢ (¬ 𝑦 = {∅} → ∃𝑥 ¬ 𝑦 = 𝑥)
6	1, 5	syl 17	. . 3 ⊢ (𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥)
7		0ex 5175	. . . 4 ⊢ ∅ ∈ V
8		eqeq2 2810	. . . . 5 ⊢ (𝑥 = ∅ → (𝑦 = 𝑥 ↔ 𝑦 = ∅))
9	8	notbid 321	. . . 4 ⊢ (𝑥 = ∅ → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = ∅))
10	7, 9	spcev 3555	. . 3 ⊢ (¬ 𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥)
11	6, 10	pm2.61i 185	. 2 ⊢ ∃𝑥 ¬ 𝑦 = 𝑥
12		exnal 1828	. . 3 ⊢ (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑦 = 𝑥)
13		eqcom 2805	. . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
14	13	albii 1821	. . 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

Syntax hints:  ¬ wn 3  ∀wal 1536   = wceq 1538  ∃wex 1781  ∅c0 4243  {csn 4525

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-sn 4526  df-pr 4528

This theorem is referenced by: (None)