Theorem regsfromunir1 37543

Description: Derivation of ax-regs 35935 from unir1 9858. (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 9839	. . . . 5 ⊢ rank:∪ (𝑅1 “ On)⟶On
2		fimass 6760	. . . . 5 ⊢ (rank:∪ (𝑅1 “ On)⟶On → (rank “ {𝑥 ∣ 𝜑}) ⊆ On)
3	1, 2	ax-mp 5	. . . 4 ⊢ (rank “ {𝑥 ∣ 𝜑}) ⊆ On
4		ffn 6739	. . . . . . . 8 ⊢ (rank:∪ (𝑅1 “ On)⟶On → rank Fn ∪ (𝑅1 “ On))
5	1, 4	ax-mp 5	. . . . . . 7 ⊢ rank Fn ∪ (𝑅1 “ On)
6		ssv 3937	. . . . . . . 8 ⊢ {𝑥 ∣ 𝜑} ⊆ V
7		regsfromunir1.1	. . . . . . . 8 ⊢ ∪ (𝑅1 “ On) = V
8	6, 7	sseqtrri 3962	. . . . . . 7 ⊢ {𝑥 ∣ 𝜑} ⊆ ∪ (𝑅1 “ On)
9		fnimaeq0 6702	. . . . . . 7 ⊢ ((rank Fn ∪ (𝑅1 “ On) ∧ {𝑥 ∣ 𝜑} ⊆ ∪ (𝑅1 “ On)) → ((rank “ {𝑥 ∣ 𝜑}) = ∅ ↔ {𝑥 ∣ 𝜑} = ∅))
10	5, 8, 9	mp2an 705	. . . . . 6 ⊢ ((rank “ {𝑥 ∣ 𝜑}) = ∅ ↔ {𝑥 ∣ 𝜑} = ∅)
11	10	necon3bii 3011	. . . . 5 ⊢ ((rank “ {𝑥 ∣ 𝜑}) ≠ ∅ ↔ {𝑥 ∣ 𝜑} ≠ ∅)
12	11	biimpri 231	. . . 4 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ → (rank “ {𝑥 ∣ 𝜑}) ≠ ∅)
13		onint 7838	. . . 4 ⊢ (((rank “ {𝑥 ∣ 𝜑}) ⊆ On ∧ (rank “ {𝑥 ∣ 𝜑}) ≠ ∅) → ∩ (rank “ {𝑥 ∣ 𝜑}) ∈ (rank “ {𝑥 ∣ 𝜑}))
14	3, 12, 13	sylancr 599	. . 3 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ → ∩ (rank “ {𝑥 ∣ 𝜑}) ∈ (rank “ {𝑥 ∣ 𝜑}))
15		abn0 4337	. . 3 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
16		fvelimab 6988	. . . 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 705	. . 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 3458	. . . . . . 7 ⊢ 𝑦 ∈ V
20	19, 7	eleqtrri 2863	. . . . . 6 ⊢ 𝑦 ∈ ∪ (𝑅1 “ On)
21		rankelb 9870	. . . . . 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 2853	. . . . . 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 7270	. . . . . . . 8 ⊢ ((rank Fn ∪ (𝑅1 “ On) ∧ {𝑥 ∣ 𝜑} ⊆ ∪ (𝑅1 “ On) ∧ 𝑧 ∈ {𝑥 ∣ 𝜑}) → (rank‘𝑧) ∈ (rank “ {𝑥 ∣ 𝜑}))
26	5, 8, 25	mp3an12 1481	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} → (rank‘𝑧) ∈ (rank “ {𝑥 ∣ 𝜑}))
27		onnmin 7846	. . . . . . 7 ⊢ (((rank “ {𝑥 ∣ 𝜑}) ⊆ On ∧ (rank‘𝑧) ∈ (rank “ {𝑥 ∣ 𝜑})) → ¬ (rank‘𝑧) ∈ ∩ (rank “ {𝑥 ∣ 𝜑}))
28	3, 26, 27	sylancr 599	. . . . . 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 1962	. . 3 ⊢ ((rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}) → ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}))
32	31	reximi 3104	. 2 ⊢ (∃𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}) → ∃𝑦 ∈ {𝑥 ∣ 𝜑}∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}))
33		df-rex 3091	. . 3 ⊢ (∃𝑦 ∈ {𝑥 ∣ 𝜑}∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}) ↔ ∃𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑})))
34		df-clab 2742	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
35		sb6 2124	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
36	34, 35	bitri 278	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
37		df-clab 2742	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
38		sb6 2124	. . . . . . . . 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 1852	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
43	36, 42	anbi12i 640	. . . 4 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑})) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
44	43	exbii 1881	. . 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))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812  [wsb 2101   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  ∪ cuni 4872  ∩ cint 4912   “ cima 5661  Oncon0 6388   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  𝑅₁cr1 9803  rankcrnk 9804

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-r1 9805  df-rank 9806

This theorem is used by: (None)