Theorem dtruALT 5317

Description: Alternate proof of dtru 5376 which requires more axioms but is shorter and may be easier to understand. Like dtruALT2 5299, it uses ax-pow 5294 rather than ax-pr 5362.

Assuming that ZF set theory is consistent, we cannot prove this theorem unless we specify that 𝑥 and 𝑦 be distinct. Specifically, Theorem spcev 3544 requires that 𝑥 must not occur in the subexpression ¬ 𝑦 = {∅} in step 4 nor in the subexpression ¬ 𝑦 = ∅ in step 9. The proof verifier will require that 𝑥 and 𝑦 be in a distinct variable group to ensure this. You can check this by deleting the $d statement in set.mm and rerunning the verifier, which will print a detailed explanation of the distinct variable violation. (Contributed by NM, 15-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dtruALT	⊢ ¬ ∀𝑥 𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem dtruALT

Step	Hyp	Ref	Expression
1		0inp0 5287	. . . 4 ⊢ (𝑦 = ∅ → ¬ 𝑦 = {∅})
2		p0ex 5313	. . . . 5 ⊢ {∅} ∈ V
3		eqeq2 2751	. . . . . 6 ⊢ (𝑥 = {∅} → (𝑦 = 𝑥 ↔ 𝑦 = {∅}))
4	3	notbid 319	. . . . 5 ⊢ (𝑥 = {∅} → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = {∅}))
5	2, 4	spcev 3544	. . . 4 ⊢ (¬ 𝑦 = {∅} → ∃𝑥 ¬ 𝑦 = 𝑥)
6	1, 5	syl 17	. . 3 ⊢ (𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥)
7		0ex 5229	. . . 4 ⊢ ∅ ∈ V
8		eqeq2 2751	. . . . 5 ⊢ (𝑥 = ∅ → (𝑦 = 𝑥 ↔ 𝑦 = ∅))
9	8	notbid 319	. . . 4 ⊢ (𝑥 = ∅ → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = ∅))
10	7, 9	spcev 3544	. . 3 ⊢ (¬ 𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥)
11	6, 10	pm2.61i 183	. 2 ⊢ ∃𝑥 ¬ 𝑦 = 𝑥
12		exnal 1834	. . 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 335	. 2 ⊢ (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
16	11, 15	mpbi 231	1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1545   = wceq 1547  ∃wex 1786  ∅c0 4261  {csn 4555

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-v 3433  df-dif 3886  df-ss 3900  df-nul 4262  df-pw 4531  df-sn 4556

This theorem is referenced by: (None)