Theorem regsfromunir1 36936

Description: Derivation of ax-regs 35458 from unir1 9781. (Contributed by Matthew House, 4-Mar-2026.)

Hypothesis

Ref	Expression
regsfromunir1.1	⊢ ∪ (𝑅1 “ On) = V

Assertion

Ref	Expression
regsfromunir1	⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))

Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem regsfromunir1

Step	Hyp	Ref	Expression
1		rankf 9762	. . . . 5 ⊢ rank:∪ (𝑅1 “ On)⟶On
2		fimass 6724	. . . . 5 ⊢ (rank:∪ (𝑅1 “ On)⟶On → (rank “ {𝑥 ∣ 𝜑}) ⊆ On)
3	1, 2	ax-mp 5	. . . 4 ⊢ (rank “ {𝑥 ∣ 𝜑}) ⊆ On
4		ffn 6703	. . . . . . . 8 ⊢ (rank:∪ (𝑅1 “ On)⟶On → rank Fn ∪ (𝑅1 “ On))
5	1, 4	ax-mp 5	. . . . . . 7 ⊢ rank Fn ∪ (𝑅1 “ On)
6		ssv 3969	. . . . . . . 8 ⊢ {𝑥 ∣ 𝜑} ⊆ V
7		regsfromunir1.1	. . . . . . . 8 ⊢ ∪ (𝑅1 “ On) = V
8	6, 7	sseqtrri 3994	. . . . . . 7 ⊢ {𝑥 ∣ 𝜑} ⊆ ∪ (𝑅1 “ On)
9		fnimaeq0 6666	. . . . . . 7 ⊢ ((rank Fn ∪ (𝑅1 “ On) ∧ {𝑥 ∣ 𝜑} ⊆ ∪ (𝑅1 “ On)) → ((rank “ {𝑥 ∣ 𝜑}) = ∅ ↔ {𝑥 ∣ 𝜑} = ∅))
10	5, 8, 9	mp2an 704	. . . . . 6 ⊢ ((rank “ {𝑥 ∣ 𝜑}) = ∅ ↔ {𝑥 ∣ 𝜑} = ∅)
11	10	necon3bii 3016	. . . . 5 ⊢ ((rank “ {𝑥 ∣ 𝜑}) ≠ ∅ ↔ {𝑥 ∣ 𝜑} ≠ ∅)
12	11	biimpri 231	. . . 4 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ → (rank “ {𝑥 ∣ 𝜑}) ≠ ∅)
13		onint 7785	. . . 4 ⊢ (((rank “ {𝑥 ∣ 𝜑}) ⊆ On ∧ (rank “ {𝑥 ∣ 𝜑}) ≠ ∅) → ∩ (rank “ {𝑥 ∣ 𝜑}) ∈ (rank “ {𝑥 ∣ 𝜑}))
14	3, 12, 13	sylancr 598	. . 3 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ → ∩ (rank “ {𝑥 ∣ 𝜑}) ∈ (rank “ {𝑥 ∣ 𝜑}))
15		abn0 4347	. . 3 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
16		fvelimab 6951	. . . 4 ⊢ ((rank Fn ∪ (𝑅1 “ On) ∧ {𝑥 ∣ 𝜑} ⊆ ∪ (𝑅1 “ On)) → (∩ (rank “ {𝑥 ∣ 𝜑}) ∈ (rank “ {𝑥 ∣ 𝜑}) ↔ ∃𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑})))
17	5, 8, 16	mp2an 704	. . 3 ⊢ (∩ (rank “ {𝑥 ∣ 𝜑}) ∈ (rank “ {𝑥 ∣ 𝜑}) ↔ ∃𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}))
18	14, 15, 17	3imtr3i 294	. 2 ⊢ (∃𝑥𝜑 → ∃𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}))
19		vex 3467	. . . . . . 7 ⊢ 𝑦 ∈ V
20	19, 7	eleqtrri 2868	. . . . . 6 ⊢ 𝑦 ∈ ∪ (𝑅1 “ On)
21		rankelb 9792	. . . . . 6 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) → (𝑧 ∈ 𝑦 → (rank‘𝑧) ∈ (rank‘𝑦)))
22	20, 21	ax-mp 5	. . . . 5 ⊢ (𝑧 ∈ 𝑦 → (rank‘𝑧) ∈ (rank‘𝑦))
23		eleq2 2858	. . . . . 6 ⊢ ((rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}) → ((rank‘𝑧) ∈ (rank‘𝑦) ↔ (rank‘𝑧) ∈ ∩ (rank “ {𝑥 ∣ 𝜑})))
24	23	biimpd 232	. . . . 5 ⊢ ((rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}) → ((rank‘𝑧) ∈ (rank‘𝑦) → (rank‘𝑧) ∈ ∩ (rank “ {𝑥 ∣ 𝜑})))
25		fnfvima 7229	. . . . . . . 8 ⊢ ((rank Fn ∪ (𝑅1 “ On) ∧ {𝑥 ∣ 𝜑} ⊆ ∪ (𝑅1 “ On) ∧ 𝑧 ∈ {𝑥 ∣ 𝜑}) → (rank‘𝑧) ∈ (rank “ {𝑥 ∣ 𝜑}))
26	5, 8, 25	mp3an12 1477	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} → (rank‘𝑧) ∈ (rank “ {𝑥 ∣ 𝜑}))
27		onnmin 7793	. . . . . . 7 ⊢ (((rank “ {𝑥 ∣ 𝜑}) ⊆ On ∧ (rank‘𝑧) ∈ (rank “ {𝑥 ∣ 𝜑})) → ¬ (rank‘𝑧) ∈ ∩ (rank “ {𝑥 ∣ 𝜑}))
28	3, 26, 27	sylancr 598	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} → ¬ (rank‘𝑧) ∈ ∩ (rank “ {𝑥 ∣ 𝜑}))
29	28	con2i 140	. . . . 5 ⊢ ((rank‘𝑧) ∈ ∩ (rank “ {𝑥 ∣ 𝜑}) → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑})
30	22, 24, 29	syl56 37	. . . 4 ⊢ ((rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}) → (𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}))
31	30	alrimiv 1954	. . 3 ⊢ ((rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}) → ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}))
32	31	reximi 3109	. 2 ⊢ (∃𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}) → ∃𝑦 ∈ {𝑥 ∣ 𝜑}∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}))
33		df-rex 3096	. . 3 ⊢ (∃𝑦 ∈ {𝑥 ∣ 𝜑}∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}) ↔ ∃𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑})))
34		df-clab 2748	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
35		sb6 2125	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
36	34, 35	bitri 278	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
37		df-clab 2748	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
38		sb6 2125	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑))
39	37, 38	bitri 278	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑))
40	39	notbii 323	. . . . . . 7 ⊢ (¬ 𝑧 ∈ {𝑥 ∣ 𝜑} ↔ ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))
41	40	imbi2i 339	. . . . . 6 ⊢ ((𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}) ↔ (𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
42	41	albii 1846	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
43	36, 42	anbi12i 639	. . . 4 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑})) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
44	43	exbii 1875	. . 3 ⊢ (∃𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑})) ↔ ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
45	33, 44	sylbb 222	. 2 ⊢ (∃𝑦 ∈ {𝑥 ∣ 𝜑}∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}) → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
46	18, 32, 45	3syl 19	1 ⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565   = wceq 1567  ∃wex 1806  [wsb 2097   ∈ wcel 2149  {cab 2747   ≠ wne 2964  ∃wrex 3095  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  ∪ cuni 4873  ∩ cint 4913   “ cima 5662  Oncon0 6357   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  𝑅₁cr1 9730  rankcrnk 9731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7411  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-r1 9732  df-rank 9733

This theorem is referenced by: (None)