pwfseqlem5 - Metamath Proof Explorer

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Theorem pwfseqlem5 10775

Description: Lemma for pwfseq 10776. Although in some ways pwfseqlem4 10774 is the "main" part of the proof, one last aspect which makes up a remark in the original text is by far the hardest part to formalize. The main proof relies on the existence of an injection 𝐾 from the set of finite sequences on an infinite set 𝑥 to 𝑥. Now this alone would not be difficult to prove; this is mostly the claim of fseqen 10133. However, what is needed for the proof is a canonical injection on these sets, so we have to start from scratch pulling together explicit bijections from the lemmas.

If one attempts such a program, it will mostly go through, but there is one key step which is inherently nonconstructive, namely the proof of infxpen 10120. The resolution is not obvious, but it turns out that reversing an infinite ordinal's Cantor normal form absorbs all the non-leading terms (cnfcom3c 9734), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 10124). (Contributed by Mario Carneiro, 31-May-2015.)

**Hypotheses**
Ref	Expression
pwfseqlem5.g	⊢ (𝜑 → 𝐺:𝒫 𝐴⟶1-1∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛))
pwfseqlem5.x	⊢ (𝜑 → 𝑋 ⊆ 𝐴)
pwfseqlem5.h	⊢ (𝜑 → 𝐻:ω⟶1-1onto𝑋)
pwfseqlem5.ps	⊢ (𝜓 ↔ ((𝑡 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡))
pwfseqlem5.n	⊢ (𝜑 → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁‘𝑏):(𝑏 × 𝑏)⟶1-1onto𝑏))
pwfseqlem5.o	⊢ 𝑂 = OrdIso(𝑟, 𝑡)
pwfseqlem5.t	⊢ 𝑇 = (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩)
pwfseqlem5.p	⊢ 𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ^◡𝑇)
pwfseqlem5.s	⊢ 𝑆 = seq_ω((𝑘 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑡 ↑_m suc 𝑘) ↦ ((𝑓‘(𝑥 ↾ 𝑘))𝑃(𝑥‘𝑘)))), {⟨∅, (𝑂‘∅)⟩})
pwfseqlem5.q	⊢ 𝑄 = (𝑦 ∈ ∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛) ↦ ⟨dom 𝑦, ((𝑆‘dom 𝑦)‘𝑦)⟩)
pwfseqlem5.i	⊢ 𝐼 = (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩)
pwfseqlem5.k	⊢ 𝐾 = ((𝑃 ∘ 𝐼) ∘ 𝑄)

**Assertion**
Ref	Expression
pwfseqlem5	⊢ ¬ 𝜑

Distinct variable groups: 𝑛,𝑏,𝐺 𝑟,𝑏,𝑡,𝐻 𝑓,𝑘,𝑥,𝑃 𝑓,𝑏,𝑘,𝑢,𝑣,𝑥,𝑦,𝑛,𝑟,𝑡 𝜑,𝑏,𝑘,𝑛,𝑟,𝑡,𝑥,𝑦 𝐾,𝑏,𝑛 𝑁,𝑏 𝜓,𝑘,𝑛,𝑥,𝑦 𝑆,𝑛,𝑦 𝐴,𝑏,𝑛,𝑟,𝑡 𝑂,𝑏,𝑢,𝑣,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑣, 𝑢, 𝑓) 𝜓(𝑣, 𝑢, 𝑡, 𝑓, 𝑟, 𝑏) 𝐴(𝑥, 𝑦, 𝑣, 𝑢, 𝑓, 𝑘) 𝑃(𝑦, 𝑣, 𝑢, 𝑡, 𝑛, 𝑟, 𝑏) 𝑄(𝑥, 𝑦, 𝑣, 𝑢, 𝑡, 𝑓, 𝑘, 𝑛, 𝑟, 𝑏) 𝑆(𝑥, 𝑣, 𝑢, 𝑡, 𝑓, 𝑘, 𝑟, 𝑏) 𝑇(𝑥, 𝑦, 𝑣, 𝑢, 𝑡, 𝑓, 𝑘, 𝑛, 𝑟, 𝑏) 𝐺(𝑥, 𝑦, 𝑣, 𝑢, 𝑡, 𝑓, 𝑘, 𝑟) 𝐻(𝑥, 𝑦, 𝑣, 𝑢, 𝑓, 𝑘, 𝑛) 𝐼(𝑥, 𝑦, 𝑣, 𝑢, 𝑡, 𝑓, 𝑘, 𝑛, 𝑟, 𝑏) 𝐾(𝑥, 𝑦, 𝑣, 𝑢, 𝑡, 𝑓, 𝑘, 𝑟) 𝑁(𝑥, 𝑦, 𝑣, 𝑢, 𝑡, 𝑓, 𝑘, 𝑛, 𝑟) 𝑂(𝑡, 𝑓, 𝑘, 𝑛, 𝑟) 𝑋(𝑥, 𝑦, 𝑣, 𝑢, 𝑡, 𝑓, 𝑘, 𝑛, 𝑟, 𝑏)

Proof of Theorem pwfseqlem5 Dummy variables 𝑎 𝑐 𝑑 𝑖 𝑗 𝑚 𝑠 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwfseqlem5.g	. 2 ⊢ (𝜑 → 𝐺:𝒫 𝐴⟶1-1∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛))
2		pwfseqlem5.x	. 2 ⊢ (𝜑 → 𝑋 ⊆ 𝐴)
3		pwfseqlem5.h	. 2 ⊢ (𝜑 → 𝐻:ω⟶1-1onto𝑋)
4		pwfseqlem5.ps	. 2 ⊢ (𝜓 ↔ ((𝑡 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡))
5		vex 3458	. . . . . . . . . . 11 ⊢ 𝑡 ∈ V
6		simprl3 1239	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑡 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → 𝑟 We 𝑡)
7	4, 6	sylan2b 606	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝑟 We 𝑡)
8		pwfseqlem5.o	. . . . . . . . . . . 12 ⊢ 𝑂 = OrdIso(𝑟, 𝑡)
9	8	oiiso 9558	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ V ∧ 𝑟 We 𝑡) → 𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡))
10	5, 7, 9	sylancr 599	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → 𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡))
11		isof1o 7354	. . . . . . . . . 10 ⊢ (𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡) → 𝑂:dom 𝑂⟶1-1onto𝑡)
12	10, 11	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑂:dom 𝑂⟶1-1onto𝑡)
13		cardom 10094	. . . . . . . . . . . 12 ⊢ (card‘ω) = ω
14		simprr 785	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑡 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → ω ≼ 𝑡)
15	4, 14	sylan2b 606	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → ω ≼ 𝑡)
16	8	oien 9559	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ V ∧ 𝑟 We 𝑡) → dom 𝑂 ≈ 𝑡)
17	5, 7, 16	sylancr 599	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ≈ 𝑡)
18	17	ensymd 9060	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ≈ dom 𝑂)
19		domentr 9068	. . . . . . . . . . . . . 14 ⊢ ((ω ≼ 𝑡 ∧ 𝑡 ≈ dom 𝑂) → ω ≼ dom 𝑂)
20	15, 18, 19	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ω ≼ dom 𝑂)
21		omelon 9674	. . . . . . . . . . . . . . 15 ⊢ ω ∈ On
22		onenon 10057	. . . . . . . . . . . . . . 15 ⊢ (ω ∈ On → ω ∈ dom card)
23	21, 22	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ω ∈ dom card
24	8	oion 9557	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ V → dom 𝑂 ∈ On)
25	24	elv 3459	. . . . . . . . . . . . . . 15 ⊢ dom 𝑂 ∈ On
26		onenon 10057	. . . . . . . . . . . . . . 15 ⊢ (dom 𝑂 ∈ On → dom 𝑂 ∈ dom card)
27	25, 26	mp1i 14	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ∈ dom card)
28		carddom2 10085	. . . . . . . . . . . . . 14 ⊢ ((ω ∈ dom card ∧ dom 𝑂 ∈ dom card) → ((card‘ω) ⊆ (card‘dom 𝑂) ↔ ω ≼ dom 𝑂))
29	23, 27, 28	sylancr 599	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ((card‘ω) ⊆ (card‘dom 𝑂) ↔ ω ≼ dom 𝑂))
30	20, 29	mpbird 260	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (card‘ω) ⊆ (card‘dom 𝑂))
31	13, 30	eqsstrrid 3952	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → ω ⊆ (card‘dom 𝑂))
32		cardonle 10065	. . . . . . . . . . . 12 ⊢ (dom 𝑂 ∈ On → (card‘dom 𝑂) ⊆ dom 𝑂)
33	25, 32	mp1i 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (card‘dom 𝑂) ⊆ dom 𝑂)
34	31, 33	sstrd 3923	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → ω ⊆ dom 𝑂)
35		sseq2 3939	. . . . . . . . . . . 12 ⊢ (𝑏 = dom 𝑂 → (ω ⊆ 𝑏 ↔ ω ⊆ dom 𝑂))
36		fveq2 6908	. . . . . . . . . . . . . 14 ⊢ (𝑏 = dom 𝑂 → (𝑁‘𝑏) = (𝑁‘dom 𝑂))
37	36	f1oeq1d 6841	. . . . . . . . . . . . 13 ⊢ (𝑏 = dom 𝑂 → ((𝑁‘𝑏):(𝑏 × 𝑏)⟶1-1onto𝑏 ↔ (𝑁‘dom 𝑂):(𝑏 × 𝑏)⟶1-1onto𝑏))
38		xpeq12 5680	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = dom 𝑂 ∧ 𝑏 = dom 𝑂) → (𝑏 × 𝑏) = (dom 𝑂 × dom 𝑂))
39	38	anidms 577	. . . . . . . . . . . . . 14 ⊢ (𝑏 = dom 𝑂 → (𝑏 × 𝑏) = (dom 𝑂 × dom 𝑂))
40	39	f1oeq2d 6842	. . . . . . . . . . . . 13 ⊢ (𝑏 = dom 𝑂 → ((𝑁‘dom 𝑂):(𝑏 × 𝑏)⟶1-1onto𝑏 ↔ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)⟶1-1onto𝑏))
41		f1oeq3 6836	. . . . . . . . . . . . 13 ⊢ (𝑏 = dom 𝑂 → ((𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)⟶1-1onto𝑏 ↔ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)⟶1-1ontodom 𝑂))
42	37, 40, 41	3bitrd 308	. . . . . . . . . . . 12 ⊢ (𝑏 = dom 𝑂 → ((𝑁‘𝑏):(𝑏 × 𝑏)⟶1-1onto𝑏 ↔ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)⟶1-1ontodom 𝑂))
43	35, 42	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑏 = dom 𝑂 → ((ω ⊆ 𝑏 → (𝑁‘𝑏):(𝑏 × 𝑏)⟶1-1onto𝑏) ↔ (ω ⊆ dom 𝑂 → (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)⟶1-1ontodom 𝑂)))
44		pwfseqlem5.n	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁‘𝑏):(𝑏 × 𝑏)⟶1-1onto𝑏))
45	44	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁‘𝑏):(𝑏 × 𝑏)⟶1-1onto𝑏))
46	25	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ∈ On)
47	1	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → 𝐺:𝒫 𝐴⟶1-1∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛))
48		omex 9671	. . . . . . . . . . . . . . . . . 18 ⊢ ω ∈ V
49		ovex 7476	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ↑_m 𝑛) ∈ V
50	48, 49	iunex 8004	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∈ V
51		f1dmex 7993	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺:𝒫 𝐴⟶1-1∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∧ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∈ V) → 𝒫 𝐴 ∈ V)
52	47, 50, 51	sylancl 598	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → 𝒫 𝐴 ∈ V)
53		pwexb 7804	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
54	52, 53	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ V)
55		simprl1 1237	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑡 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → 𝑡 ⊆ 𝐴)
56	4, 55	sylan2b 606	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ⊆ 𝐴)
57		ssdomg 9055	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ V → (𝑡 ⊆ 𝐴 → 𝑡 ≼ 𝐴))
58	54, 56, 57	sylc 66	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ≼ 𝐴)
59		canth2g 9178	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
60		sdomdom 9035	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ≺ 𝒫 𝐴 → 𝐴 ≼ 𝒫 𝐴)
61	54, 59, 60	3syl 19	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ≼ 𝒫 𝐴)
62		domtr 9062	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐴) → 𝑡 ≼ 𝒫 𝐴)
63	58, 61, 62	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ≼ 𝒫 𝐴)
64		endomtr 9067	. . . . . . . . . . . . 13 ⊢ ((dom 𝑂 ≈ 𝑡 ∧ 𝑡 ≼ 𝒫 𝐴) → dom 𝑂 ≼ 𝒫 𝐴)
65	17, 63, 64	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ≼ 𝒫 𝐴)
66		elharval 9582	. . . . . . . . . . . 12 ⊢ (dom 𝑂 ∈ (har‘𝒫 𝐴) ↔ (dom 𝑂 ∈ On ∧ dom 𝑂 ≼ 𝒫 𝐴))
67	46, 65, 66	sylanbrc 595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ∈ (har‘𝒫 𝐴))
68	43, 45, 67	rspcdva 3581	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (ω ⊆ dom 𝑂 → (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)⟶1-1ontodom 𝑂))
69	34, 68	mpd 16	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)⟶1-1ontodom 𝑂)
70		f1oco 6870	. . . . . . . . 9 ⊢ ((𝑂:dom 𝑂⟶1-1onto𝑡 ∧ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)⟶1-1ontodom 𝑂) → (𝑂 ∘ (𝑁‘dom 𝑂)):(dom 𝑂 × dom 𝑂)⟶1-1onto𝑡)
71	12, 69, 70	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 ∘ (𝑁‘dom 𝑂)):(dom 𝑂 × dom 𝑂)⟶1-1onto𝑡)
72		f1of 6846	. . . . . . . . . . . . . . 15 ⊢ (𝑂:dom 𝑂⟶1-1onto𝑡 → 𝑂:dom 𝑂⟶𝑡)
73	12, 72	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑂:dom 𝑂⟶𝑡)
74	73	feqmptd 6976	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑂 = (𝑢 ∈ dom 𝑂 ↦ (𝑂‘𝑢)))
75	74	f1oeq1d 6841	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (𝑂:dom 𝑂⟶1-1onto𝑡 ↔ (𝑢 ∈ dom 𝑂 ↦ (𝑂‘𝑢)):dom 𝑂⟶1-1onto𝑡))
76	12, 75	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑢 ∈ dom 𝑂 ↦ (𝑂‘𝑢)):dom 𝑂⟶1-1onto𝑡)
77	73	feqmptd 6976	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑂 = (𝑣 ∈ dom 𝑂 ↦ (𝑂‘𝑣)))
78	77	f1oeq1d 6841	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (𝑂:dom 𝑂⟶1-1onto𝑡 ↔ (𝑣 ∈ dom 𝑂 ↦ (𝑂‘𝑣)):dom 𝑂⟶1-1onto𝑡))
79	12, 78	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑣 ∈ dom 𝑂 ↦ (𝑂‘𝑣)):dom 𝑂⟶1-1onto𝑡)
80	76, 79	xpf1o 9186	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩):(dom 𝑂 × dom 𝑂)⟶1-1onto(𝑡 × 𝑡))
81		pwfseqlem5.t	. . . . . . . . . . 11 ⊢ 𝑇 = (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩)
82		f1oeq1 6834	. . . . . . . . . . 11 ⊢ (𝑇 = (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩) → (𝑇:(dom 𝑂 × dom 𝑂)⟶1-1onto(𝑡 × 𝑡) ↔ (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩):(dom 𝑂 × dom 𝑂)⟶1-1onto(𝑡 × 𝑡)))
83	81, 82	ax-mp 5	. . . . . . . . . 10 ⊢ (𝑇:(dom 𝑂 × dom 𝑂)⟶1-1onto(𝑡 × 𝑡) ↔ (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩):(dom 𝑂 × dom 𝑂)⟶1-1onto(𝑡 × 𝑡))
84	80, 83	sylibr 237	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑇:(dom 𝑂 × dom 𝑂)⟶1-1onto(𝑡 × 𝑡))
85		f1ocnv 6859	. . . . . . . . 9 ⊢ (𝑇:(dom 𝑂 × dom 𝑂)⟶1-1onto(𝑡 × 𝑡) → ^◡𝑇:(𝑡 × 𝑡)⟶1-1onto(dom 𝑂 × dom 𝑂))
86	84, 85	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → ^◡𝑇:(𝑡 × 𝑡)⟶1-1onto(dom 𝑂 × dom 𝑂))
87		f1oco 6870	. . . . . . . 8 ⊢ (((𝑂 ∘ (𝑁‘dom 𝑂)):(dom 𝑂 × dom 𝑂)⟶1-1onto𝑡 ∧ ^◡𝑇:(𝑡 × 𝑡)⟶1-1onto(dom 𝑂 × dom 𝑂)) → ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ^◡𝑇):(𝑡 × 𝑡)⟶1-1onto𝑡)
88	71, 86, 87	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ^◡𝑇):(𝑡 × 𝑡)⟶1-1onto𝑡)
89		pwfseqlem5.p	. . . . . . . 8 ⊢ 𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ^◡𝑇)
90		f1oeq1 6834	. . . . . . . 8 ⊢ (𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ^◡𝑇) → (𝑃:(𝑡 × 𝑡)⟶1-1onto𝑡 ↔ ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ^◡𝑇):(𝑡 × 𝑡)⟶1-1onto𝑡))
91	89, 90	ax-mp 5	. . . . . . 7 ⊢ (𝑃:(𝑡 × 𝑡)⟶1-1onto𝑡 ↔ ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ^◡𝑇):(𝑡 × 𝑡)⟶1-1onto𝑡)
92	88, 91	sylibr 237	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑃:(𝑡 × 𝑡)⟶1-1onto𝑡)
93		f1of1 6845	. . . . . 6 ⊢ (𝑃:(𝑡 × 𝑡)⟶1-1onto𝑡 → 𝑃:(𝑡 × 𝑡)⟶1-1𝑡)
94	92, 93	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑃:(𝑡 × 𝑡)⟶1-1𝑡)
95		f1of1 6845	. . . . . . . . . . . . 13 ⊢ (𝑂:dom 𝑂⟶1-1onto𝑡 → 𝑂:dom 𝑂⟶1-1𝑡)
96	12, 95	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 𝑂:dom 𝑂⟶1-1𝑡)
97		f1ssres 6809	. . . . . . . . . . . 12 ⊢ ((𝑂:dom 𝑂⟶1-1𝑡 ∧ ω ⊆ dom 𝑂) → (𝑂 ↾ ω):ω⟶1-1𝑡)
98	96, 34, 97	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 ↾ ω):ω⟶1-1𝑡)
99		f1f1orn 6858	. . . . . . . . . . 11 ⊢ ((𝑂 ↾ ω):ω⟶1-1𝑡 → (𝑂 ↾ ω):ω⟶1-1ontoran (𝑂 ↾ ω))
100	98, 99	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 ↾ ω):ω⟶1-1ontoran (𝑂 ↾ ω))
101	73, 34	feqresmpt 6977	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 ↾ ω) = (𝑥 ∈ ω ↦ (𝑂‘𝑥)))
102	101	f1oeq1d 6841	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → ((𝑂 ↾ ω):ω⟶1-1ontoran (𝑂 ↾ ω) ↔ (𝑥 ∈ ω ↦ (𝑂‘𝑥)):ω⟶1-1ontoran (𝑂 ↾ ω)))
103	100, 102	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝑥 ∈ ω ↦ (𝑂‘𝑥)):ω⟶1-1ontoran (𝑂 ↾ ω))
104		mptresid 6055	. . . . . . . . . . 11 ⊢ ( I ↾ 𝑡) = (𝑦 ∈ 𝑡 ↦ 𝑦)
105	104	eqcomi 2772	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑡 ↦ 𝑦) = ( I ↾ 𝑡)
106		f1oi 6886	. . . . . . . . . . 11 ⊢ ( I ↾ 𝑡):𝑡⟶1-1onto𝑡
107		f1oeq1 6834	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑡 ↦ 𝑦) = ( I ↾ 𝑡) → ((𝑦 ∈ 𝑡 ↦ 𝑦):𝑡⟶1-1onto𝑡 ↔ ( I ↾ 𝑡):𝑡⟶1-1onto𝑡))
108	106, 107	mpbiri 261	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑡 ↦ 𝑦) = ( I ↾ 𝑡) → (𝑦 ∈ 𝑡 ↦ 𝑦):𝑡⟶1-1onto𝑡)
109	105, 108	mp1i 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝑦 ∈ 𝑡 ↦ 𝑦):𝑡⟶1-1onto𝑡)
110	103, 109	xpf1o 9186	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩):(ω × 𝑡)⟶1-1onto(ran (𝑂 ↾ ω) × 𝑡))
111		pwfseqlem5.i	. . . . . . . . 9 ⊢ 𝐼 = (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩)
112		f1oeq1 6834	. . . . . . . . 9 ⊢ (𝐼 = (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩) → (𝐼:(ω × 𝑡)⟶1-1onto(ran (𝑂 ↾ ω) × 𝑡) ↔ (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩):(ω × 𝑡)⟶1-1onto(ran (𝑂 ↾ ω) × 𝑡)))
113	111, 112	ax-mp 5	. . . . . . . 8 ⊢ (𝐼:(ω × 𝑡)⟶1-1onto(ran (𝑂 ↾ ω) × 𝑡) ↔ (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩):(ω × 𝑡)⟶1-1onto(ran (𝑂 ↾ ω) × 𝑡))
114	110, 113	sylibr 237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐼:(ω × 𝑡)⟶1-1onto(ran (𝑂 ↾ ω) × 𝑡))
115		f1of1 6845	. . . . . . 7 ⊢ (𝐼:(ω × 𝑡)⟶1-1onto(ran (𝑂 ↾ ω) × 𝑡) → 𝐼:(ω × 𝑡)⟶1-1(ran (𝑂 ↾ ω) × 𝑡))
116	114, 115	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝐼:(ω × 𝑡)⟶1-1(ran (𝑂 ↾ ω) × 𝑡))
117		f1f 6800	. . . . . . 7 ⊢ ((𝑂 ↾ ω):ω⟶1-1𝑡 → (𝑂 ↾ ω):ω⟶𝑡)
118		frn 6739	. . . . . . 7 ⊢ ((𝑂 ↾ ω):ω⟶𝑡 → ran (𝑂 ↾ ω) ⊆ 𝑡)
119		xpss1 5674	. . . . . . 7 ⊢ (ran (𝑂 ↾ ω) ⊆ 𝑡 → (ran (𝑂 ↾ ω) × 𝑡) ⊆ (𝑡 × 𝑡))
120	98, 117, 118, 119	4syl 20	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (ran (𝑂 ↾ ω) × 𝑡) ⊆ (𝑡 × 𝑡))
121		f1ss 6807	. . . . . 6 ⊢ ((𝐼:(ω × 𝑡)⟶1-1(ran (𝑂 ↾ ω) × 𝑡) ∧ (ran (𝑂 ↾ ω) × 𝑡) ⊆ (𝑡 × 𝑡)) → 𝐼:(ω × 𝑡)⟶1-1(𝑡 × 𝑡))
122	116, 120, 121	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐼:(ω × 𝑡)⟶1-1(𝑡 × 𝑡))
123		f1co 6813	. . . . 5 ⊢ ((𝑃:(𝑡 × 𝑡)⟶1-1𝑡 ∧ 𝐼:(ω × 𝑡)⟶1-1(𝑡 × 𝑡)) → (𝑃 ∘ 𝐼):(ω × 𝑡)⟶1-1𝑡)
124	94, 122, 123	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑃 ∘ 𝐼):(ω × 𝑡)⟶1-1𝑡)
125	5	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ∈ V)
126		peano1 7924	. . . . . . . 8 ⊢ ∅ ∈ ω
127	126	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ∅ ∈ ω)
128	34, 127	sseldd 3914	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ∅ ∈ dom 𝑂)
129	73, 128	ffvelcdmd 7109	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑂‘∅) ∈ 𝑡)
130		pwfseqlem5.s	. . . . 5 ⊢ 𝑆 = seq_ω((𝑘 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑡 ↑_m suc 𝑘) ↦ ((𝑓‘(𝑥 ↾ 𝑘))𝑃(𝑥‘𝑘)))), {⟨∅, (𝑂‘∅)⟩})
131		pwfseqlem5.q	. . . . 5 ⊢ 𝑄 = (𝑦 ∈ ∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛) ↦ ⟨dom 𝑦, ((𝑆‘dom 𝑦)‘𝑦)⟩)
132	125, 129, 92, 130, 131	fseqenlem2 10131	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑄:∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)⟶1-1(ω × 𝑡))
133		f1co 6813	. . . 4 ⊢ (((𝑃 ∘ 𝐼):(ω × 𝑡)⟶1-1𝑡 ∧ 𝑄:∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)⟶1-1(ω × 𝑡)) → ((𝑃 ∘ 𝐼) ∘ 𝑄):∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)⟶1-1𝑡)
134	124, 132, 133	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝑃 ∘ 𝐼) ∘ 𝑄):∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)⟶1-1𝑡)
135		pwfseqlem5.k	. . . 4 ⊢ 𝐾 = ((𝑃 ∘ 𝐼) ∘ 𝑄)
136		f1eq1 6795	. . . 4 ⊢ (𝐾 = ((𝑃 ∘ 𝐼) ∘ 𝑄) → (𝐾:∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)⟶1-1𝑡 ↔ ((𝑃 ∘ 𝐼) ∘ 𝑄):∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)⟶1-1𝑡))
137	135, 136	ax-mp 5	. . 3 ⊢ (𝐾:∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)⟶1-1𝑡 ↔ ((𝑃 ∘ 𝐼) ∘ 𝑄):∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)⟶1-1𝑡)
138	134, 137	sylibr 237	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)⟶1-1𝑡)
139		eqid 2763	. 2 ⊢ (𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))}) = (𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})
140		eqid 2763	. 2 ⊢ (𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡}))) = (𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))
141		eqid 2763	. . 3 ⊢ {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ⊆ 𝐴 ∧ 𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚 ∈ 𝑐 [(^◡𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))} = {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ⊆ 𝐴 ∧ 𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚 ∈ 𝑐 [(^◡𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))}
142	141	fpwwe2cbv 10742	. 2 ⊢ {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ⊆ 𝐴 ∧ 𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚 ∈ 𝑐 [(^◡𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))} = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑏 ∈ 𝑎 [(^◡𝑠 “ {𝑏}) / 𝑤](𝑤(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))(𝑠 ∩ (𝑤 × 𝑤))) = 𝑏))}
143		eqid 2763	. 2 ⊢ ∪ dom {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ⊆ 𝐴 ∧ 𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚 ∈ 𝑐 [(^◡𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))} = ∪ dom {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ⊆ 𝐴 ∧ 𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚 ∈ 𝑐 [(^◡𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))}
144	1, 2, 3, 4, 138, 139, 140, 142, 143	pwfseqlem4 10774	1 ⊢ ¬ 𝜑

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∀wral 3080 {crab 3416 Vcvv 3454 [wsbc 3742 ⊆ wss 3900 ∩ cin 4092 ∅c0 4281 ifcif 4485 𝒫 cpw 4560 {csn 4588 ⟨cop 4838 ∪ cuni 4870 ∩ cint 4910 ∪ ciun 4954 class class class wbr 5106 {copab 5170 ↦ cmpt 5189 I cid 5549 E cep 5554 We wwe 5607 × cxp 5653 ^◡ccnv 5654 dom cdm 5655 ran crn 5656 ↾ cres 5657 “ cima 5658 ∘ ccom 5659 Oncon0 6380 suc csuc 6382 ⟶wf 6554 ⟶1-1wf1 6555 ⟶1-1ontowf1o 6557 ‘cfv 6558 Isom wiso 6559 (class class class)co 7443 ∈ cmpo 7445 ωcom 7901 seq_ωcseqom 8485 ↑_m cmap 8875 ≈ cen 8998 ≼ cdom 8999 ≺ csdm 9000 Fincfn 9001 OrdIsocoi 9530 harchar 9577 cardccrd 10043

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-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-inf2 9669

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-rmo 3369 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-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-seqom 8486 df-1o 8504 df-er 8745 df-map 8877 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-oi 9531 df-har 9578 df-card 10047

This theorem is used by: pwfseq 10776

W3C validator