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Theorem pwfseqlem4 10607
Description: Lemma for pwfseq 10609. Derive a final contradiction from the function 𝐹 in pwfseqlem3 10605. Applying fpwwe2 10588 to it, we get a certain maximal well-ordered subset 𝑍, but the defining property (𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍 contradicts our assumption on 𝐹, so we are reduced to the case of 𝑍 finite. This too is a contradiction, though, because 𝑍 and its preimage under (𝑊‘𝑍) are distinct sets of the same cardinality and in a subset relation, which is impossible for finite sets. (Contributed by Mario Carneiro, 31-May-2015.)
Hypotheses
Ref Expression
pwfseqlem4.g (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
pwfseqlem4.x (𝜑 → 𝑋 ⊆ 𝐴)
pwfseqlem4.h (𝜑 → 𝐻:ω–1-1-onto→𝑋)
pwfseqlem4.ps (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))
pwfseqlem4.k ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥)
pwfseqlem4.d 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})
pwfseqlem4.f 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
pwfseqlem4.w 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑏 ∈ 𝑎 [(◡𝑠 “ {𝑏}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑏))}
pwfseqlem4.z 𝑍 = ∪ dom 𝑊
Assertion
Ref Expression
pwfseqlem4 ¬ 𝜑
Distinct variable groups:   𝑛,𝑟,𝑤,𝑥,𝑧   𝐷,𝑛,𝑧   𝑎,𝑏,𝑠,𝑣,𝐹   𝑤,𝐺   𝑤,𝐾   𝑟,𝑎,𝑥,𝑧,𝐻,𝑏,𝑠,𝑣   𝑛,𝑎,𝜑,𝑏,𝑠,𝑣,𝑟,𝑥,𝑧   𝜓,𝑛,𝑧   𝐴,𝑎,𝑛,𝑟,𝑠,𝑥,𝑧   𝑊,𝑎,𝑏,𝑠,𝑣   𝑍,𝑎,𝑏,𝑠,𝑣
Allowed substitution hints:   𝜑(𝑤)   𝜓(𝑥,𝑤,𝑣,𝑠,𝑟,𝑎,𝑏)   𝐴(𝑤,𝑣,𝑏)   𝐷(𝑥,𝑤,𝑣,𝑠,𝑟,𝑎,𝑏)   𝐹(𝑥,𝑧,𝑤,𝑛,𝑟)   𝐺(𝑥,𝑧,𝑣,𝑛,𝑠,𝑟,𝑎,𝑏)   𝐻(𝑤,𝑛)   𝐾(𝑥,𝑧,𝑣,𝑛,𝑠,𝑟,𝑎,𝑏)   𝑊(𝑥,𝑧,𝑤,𝑛,𝑟)   𝑋(𝑥,𝑧,𝑤,𝑣,𝑛,𝑠,𝑟,𝑎,𝑏)   𝑍(𝑥,𝑧,𝑤,𝑛,𝑟)
Proof of Theorem pwfseqlem4
StepHypRef Expression
1 eqid 2731 . . . . . . . . . . 11 𝑍 = 𝑍
2 eqid 2731 . . . . . . . . . . 11 (𝑊‘𝑍) = (𝑊‘𝑍)
31, 2pm3.2i 471 . . . . . . . . . 10 (𝑍 = 𝑍 ∧ (𝑊‘𝑍) = (𝑊‘𝑍))
4 pwfseqlem4.w . . . . . . . . . . 11 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑏 ∈ 𝑎 [(◡𝑠 “ {𝑏}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑏))}
5 pwfseqlem4.g . . . . . . . . . . . . 13 (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
6 omex 9588 . . . . . . . . . . . . . 14 ω ∈ V
7 ovex 7395 . . . . . . . . . . . . . 14 (𝐴 ↑m 𝑛) ∈ V
86, 7iunex 7906 . . . . . . . . . . . . 13 ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∈ V
9 f1dmex 7894 . . . . . . . . . . . . 13 ((𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∈ V) → 𝒫 𝐴 ∈ V)
105, 8, 9sylancl 586 . . . . . . . . . . . 12 (𝜑 → 𝒫 𝐴 ∈ V)
11 pwexb 7705 . . . . . . . . . . . 12 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
1210, 11sylibr 233 . . . . . . . . . . 11 (𝜑 → 𝐴 ∈ V)
13 pwfseqlem4.x . . . . . . . . . . . 12 (𝜑 → 𝑋 ⊆ 𝐴)
14 pwfseqlem4.h . . . . . . . . . . . 12 (𝜑 → 𝐻:ω–1-1-onto→𝑋)
15 pwfseqlem4.ps . . . . . . . . . . . 12 (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))
16 pwfseqlem4.k . . . . . . . . . . . 12 ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥)
17 pwfseqlem4.d . . . . . . . . . . . 12 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})
18 pwfseqlem4.f . . . . . . . . . . . 12 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
195, 13, 14, 15, 16, 17, 18pwfseqlem4a 10606 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)
20 pwfseqlem4.z . . . . . . . . . . 11 𝑍 = ∪ dom 𝑊
214, 12, 19, 20fpwwe2 10588 . . . . . . . . . 10 (𝜑 → ((𝑍𝑊(𝑊‘𝑍) ∧ (𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍) ↔ (𝑍 = 𝑍 ∧ (𝑊‘𝑍) = (𝑊‘𝑍))))
223, 21mpbiri 257 . . . . . . . . 9 (𝜑 → (𝑍𝑊(𝑊‘𝑍) ∧ (𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍))
2322simprd 496 . . . . . . . 8 (𝜑 → (𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍)
2422simpld 495 . . . . . . . . . . . . 13 (𝜑 → 𝑍𝑊(𝑊‘𝑍))
254, 12fpwwe2lem2 10577 . . . . . . . . . . . . 13 (𝜑 → (𝑍𝑊(𝑊‘𝑍) ↔ ((𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍)) ∧ ((𝑊‘𝑍) We 𝑍 ∧ ∀𝑏 ∈ 𝑍 [(◡(𝑊‘𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊‘𝑍) ∩ (𝑣 × 𝑣))) = 𝑏))))
2624, 25mpbid 231 . . . . . . . . . . . 12 (𝜑 → ((𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍)) ∧ ((𝑊‘𝑍) We 𝑍 ∧ ∀𝑏 ∈ 𝑍 [(◡(𝑊‘𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊‘𝑍) ∩ (𝑣 × 𝑣))) = 𝑏)))
2726simpld 495 . . . . . . . . . . 11 (𝜑 → (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍)))
2827simpld 495 . . . . . . . . . 10 (𝜑 → 𝑍 ⊆ 𝐴)
2912, 28ssexd 5286 . . . . . . . . 9 (𝜑 → 𝑍 ∈ V)
30 sseq1 3972 . . . . . . . . . . . . . 14 (𝑎 = 𝑍 → (𝑎 ⊆ 𝐴 ↔ 𝑍 ⊆ 𝐴))
31 id 22 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑍 → 𝑎 = 𝑍)
3231sqxpeqd 5670 . . . . . . . . . . . . . . 15 (𝑎 = 𝑍 → (𝑎 × 𝑎) = (𝑍 × 𝑍))
3332sseq2d 3979 . . . . . . . . . . . . . 14 (𝑎 = 𝑍 → ((𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ↔ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍)))
34 weeq2 5627 . . . . . . . . . . . . . 14 (𝑎 = 𝑍 → ((𝑊‘𝑍) We 𝑎 ↔ (𝑊‘𝑍) We 𝑍))
3530, 33, 343anbi123d 1436 . . . . . . . . . . . . 13 (𝑎 = 𝑍 → ((𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎) ↔ (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍)))
3635anbi2d 629 . . . . . . . . . . . 12 (𝑎 = 𝑍 → ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)) ↔ (𝜑 ∧ (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍))))
37 id 22 . . . . . . . . . . . . . . . 16 ((𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍) → (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍))
38373expa 1118 . . . . . . . . . . . . . . 15 (((𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍)) ∧ (𝑊‘𝑍) We 𝑍) → (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍))
3938adantrr 715 . . . . . . . . . . . . . 14 (((𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍)) ∧ ((𝑊‘𝑍) We 𝑍 ∧ ∀𝑏 ∈ 𝑍 [(◡(𝑊‘𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊‘𝑍) ∩ (𝑣 × 𝑣))) = 𝑏)) → (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍))
4026, 39syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍))
4140pm4.71i 560 . . . . . . . . . . . 12 (𝜑 ↔ (𝜑 ∧ (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍)))
4236, 41bitr4di 288 . . . . . . . . . . 11 (𝑎 = 𝑍 → ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)) ↔ 𝜑))
43 oveq1 7369 . . . . . . . . . . . . 13 (𝑎 = 𝑍 → (𝑎𝐹(𝑊‘𝑍)) = (𝑍𝐹(𝑊‘𝑍)))
4443, 31eleq12d 2826 . . . . . . . . . . . 12 (𝑎 = 𝑍 → ((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 ↔ (𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍))
45 breq1 5113 . . . . . . . . . . . 12 (𝑎 = 𝑍 → (𝑎 ≺ ω ↔ 𝑍 ≺ ω))
4644, 45imbi12d 344 . . . . . . . . . . 11 (𝑎 = 𝑍 → (((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 → 𝑎 ≺ ω) ↔ ((𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍 → 𝑍 ≺ ω)))
4742, 46imbi12d 344 . . . . . . . . . 10 (𝑎 = 𝑍 → (((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)) → ((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 → 𝑎 ≺ ω)) ↔ (𝜑 → ((𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍 → 𝑍 ≺ ω))))
48 fvex 6860 . . . . . . . . . . 11 (𝑊‘𝑍) ∈ V
49 sseq1 3972 . . . . . . . . . . . . . 14 (𝑠 = (𝑊‘𝑍) → (𝑠 ⊆ (𝑎 × 𝑎) ↔ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎)))
50 weeq1 5626 . . . . . . . . . . . . . 14 (𝑠 = (𝑊‘𝑍) → (𝑠 We 𝑎 ↔ (𝑊‘𝑍) We 𝑎))
5149, 503anbi23d 1439 . . . . . . . . . . . . 13 (𝑠 = (𝑊‘𝑍) → ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ↔ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)))
5251anbi2d 629 . . . . . . . . . . . 12 (𝑠 = (𝑊‘𝑍) → ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) ↔ (𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎))))
53 oveq2 7370 . . . . . . . . . . . . . 14 (𝑠 = (𝑊‘𝑍) → (𝑎𝐹𝑠) = (𝑎𝐹(𝑊‘𝑍)))
5453eleq1d 2817 . . . . . . . . . . . . 13 (𝑠 = (𝑊‘𝑍) → ((𝑎𝐹𝑠) ∈ 𝑎 ↔ (𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎))
5554imbi1d 341 . . . . . . . . . . . 12 (𝑠 = (𝑊‘𝑍) → (((𝑎𝐹𝑠) ∈ 𝑎 → 𝑎 ≺ ω) ↔ ((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 → 𝑎 ≺ ω)))
5652, 55imbi12d 344 . . . . . . . . . . 11 (𝑠 = (𝑊‘𝑍) → (((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ((𝑎𝐹𝑠) ∈ 𝑎 → 𝑎 ≺ ω)) ↔ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)) → ((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 → 𝑎 ≺ ω))))
57 omelon 9591 . . . . . . . . . . . . . . 15 ω ∈ On
58 onenon 9894 . . . . . . . . . . . . . . 15 (ω ∈ On → ω ∈ dom card)
5957, 58ax-mp 5 . . . . . . . . . . . . . 14 ω ∈ dom card
60 simpr3 1196 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑠 We 𝑎)
616019.8ad 2175 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ∃𝑠 𝑠 We 𝑎)
62 ween 9980 . . . . . . . . . . . . . . 15 (𝑎 ∈ dom card ↔ ∃𝑠 𝑠 We 𝑎)
6361, 62sylibr 233 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑎 ∈ dom card)
64 domtri2 9934 . . . . . . . . . . . . . 14 ((ω ∈ dom card ∧ 𝑎 ∈ dom card) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
6559, 63, 64sylancr 587 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
66 nfv 1917 . . . . . . . . . . . . . . . . 17 Ⅎ𝑟(𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))
67 nfcv 2902 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑟𝑎
68 nfmpo2 7443 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
6918, 68nfcxfr 2900 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑟𝐹
70 nfcv 2902 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑟𝑠
7167, 69, 70nfov 7392 . . . . . . . . . . . . . . . . . 18 Ⅎ𝑟(𝑎𝐹𝑠)
7271nfel1 2918 . . . . . . . . . . . . . . . . 17 Ⅎ𝑟(𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎)
7366, 72nfim 1899 . . . . . . . . . . . . . . . 16 Ⅎ𝑟((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎))
74 sseq1 3972 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = 𝑠 → (𝑟 ⊆ (𝑎 × 𝑎) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
75 weeq1 5626 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = 𝑠 → (𝑟 We 𝑎 ↔ 𝑠 We 𝑎))
7674, 753anbi23d 1439 . . . . . . . . . . . . . . . . . . 19 (𝑟 = 𝑠 → ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)))
7776anbi1d 630 . . . . . . . . . . . . . . . . . 18 (𝑟 = 𝑠 → (((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎) ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)))
7877anbi2d 629 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑠 → ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) ↔ (𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))))
79 oveq2 7370 . . . . . . . . . . . . . . . . . 18 (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
8079eleq1d 2817 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑠 → ((𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎) ↔ (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎)))
8178, 80imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑠 → (((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎)) ↔ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎))))
82 nfv 1917 . . . . . . . . . . . . . . . . . 18 Ⅎ𝑥(𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))
83 nfcv 2902 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑥𝑎
84 nfmpo1 7442 . . . . . . . . . . . . . . . . . . . . 21 Ⅎ𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
8518, 84nfcxfr 2900 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑥𝐹
86 nfcv 2902 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑥𝑟
8783, 85, 86nfov 7392 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑥(𝑎𝐹𝑟)
8887nfel1 2918 . . . . . . . . . . . . . . . . . 18 Ⅎ𝑥(𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎)
8982, 88nfim 1899 . . . . . . . . . . . . . . . . 17 Ⅎ𝑥((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎))
90 sseq1 3972 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → (𝑥 ⊆ 𝐴 ↔ 𝑎 ⊆ 𝐴))
91 xpeq12 5663 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 = 𝑎 ∧ 𝑥 = 𝑎) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
9291anidms 567 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑎 → (𝑥 × 𝑥) = (𝑎 × 𝑎))
9392sseq2d 3979 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑎 × 𝑎)))
94 weeq2 5627 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → (𝑟 We 𝑥 ↔ 𝑟 We 𝑎))
9590, 93, 943anbi123d 1436 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎)))
96 breq2 5114 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → (ω ≼ 𝑥 ↔ ω ≼ 𝑎))
9795, 96anbi12d 631 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥) ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
9815, 97bitrid 282 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝜓 ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
9998anbi2d 629 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))))
100 oveq1 7369 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
101 difeq2 4081 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑎))
102100, 101eleq12d 2826 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥) ↔ (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎)))
10399, 102imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (((𝜑 ∧ 𝜓) → (𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥)) ↔ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎))))
1045, 13, 14, 15, 16, 17, 18pwfseqlem3 10605 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥))
10589, 103, 104chvarfv 2233 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎))
10673, 81, 105chvarfv 2233 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎))
107106eldifbd 3926 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → ¬ (𝑎𝐹𝑠) ∈ 𝑎)
108107expr 457 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 → ¬ (𝑎𝐹𝑠) ∈ 𝑎))
10965, 108sylbird 259 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (¬ 𝑎 ≺ ω → ¬ (𝑎𝐹𝑠) ∈ 𝑎))
110109con4d 115 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ((𝑎𝐹𝑠) ∈ 𝑎 → 𝑎 ≺ ω))
11148, 56, 110vtocl 3519 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)) → ((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 → 𝑎 ≺ ω))
11247, 111vtoclg 3526 . . . . . . . . 9 (𝑍 ∈ V → (𝜑 → ((𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍 → 𝑍 ≺ ω)))
11329, 112mpcom 38 . . . . . . . 8 (𝜑 → ((𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍 → 𝑍 ≺ ω))
11423, 113mpd 15 . . . . . . 7 (𝜑 → 𝑍 ≺ ω)
115 isfinite 9597 . . . . . . 7 (𝑍 ∈ Fin ↔ 𝑍 ≺ ω)
116114, 115sylibr 233 . . . . . 6 (𝜑 → 𝑍 ∈ Fin)
1175, 13, 14, 15, 16, 17, 18pwfseqlem2 10604 . . . . . 6 ((𝑍 ∈ Fin ∧ (𝑊‘𝑍) ∈ V) → (𝑍𝐹(𝑊‘𝑍)) = (𝐻‘(card‘𝑍)))
118116, 48, 117sylancl 586 . . . . 5 (𝜑 → (𝑍𝐹(𝑊‘𝑍)) = (𝐻‘(card‘𝑍)))
119118, 23eqeltrrd 2833 . . . 4 (𝜑 → (𝐻‘(card‘𝑍)) ∈ 𝑍)
1204, 12, 24fpwwe2lem3 10578 . . . . . . . . . 10 ((𝜑 ∧ (𝐻‘(card‘𝑍)) ∈ 𝑍) → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊‘𝑍) ∩ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) × (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘𝑍)))
121119, 120mpdan 685 . . . . . . . . 9 (𝜑 → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊‘𝑍) ∩ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) × (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘𝑍)))
122 cnvimass 6038 . . . . . . . . . . . 12 (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ dom (𝑊‘𝑍)
12327simprd 496 . . . . . . . . . . . . . 14 (𝜑 → (𝑊‘𝑍) ⊆ (𝑍 × 𝑍))
124 dmss 5863 . . . . . . . . . . . . . 14 ((𝑊‘𝑍) ⊆ (𝑍 × 𝑍) → dom (𝑊‘𝑍) ⊆ dom (𝑍 × 𝑍))
125123, 124syl 17 . . . . . . . . . . . . 13 (𝜑 → dom (𝑊‘𝑍) ⊆ dom (𝑍 × 𝑍))
126 dmxpss 6128 . . . . . . . . . . . . 13 dom (𝑍 × 𝑍) ⊆ 𝑍
127125, 126sstrdi 3959 . . . . . . . . . . . 12 (𝜑 → dom (𝑊‘𝑍) ⊆ 𝑍)
128122, 127sstrid 3958 . . . . . . . . . . 11 (𝜑 → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍)
129116, 128ssfid 9218 . . . . . . . . . 10 (𝜑 → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin)
13048inex1 5279 . . . . . . . . . 10 ((𝑊‘𝑍) ∩ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) × (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))) ∈ V
1315, 13, 14, 15, 16, 17, 18pwfseqlem2 10604 . . . . . . . . . 10 (((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin ∧ ((𝑊‘𝑍) ∩ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) × (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))) ∈ V) → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊‘𝑍) ∩ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) × (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))))
132129, 130, 131sylancl 586 . . . . . . . . 9 (𝜑 → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊‘𝑍) ∩ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) × (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))))
133121, 132eqtr3d 2773 . . . . . . . 8 (𝜑 → (𝐻‘(card‘𝑍)) = (𝐻‘(card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))))
134 f1of1 6788 . . . . . . . . . 10 (𝐻:ω–1-1-onto→𝑋 → 𝐻:ω–1-1→𝑋)
13514, 134syl 17 . . . . . . . . 9 (𝜑 → 𝐻:ω–1-1→𝑋)
136 ficardom 9906 . . . . . . . . . 10 (𝑍 ∈ Fin → (card‘𝑍) ∈ ω)
137116, 136syl 17 . . . . . . . . 9 (𝜑 → (card‘𝑍) ∈ ω)
138 ficardom 9906 . . . . . . . . . 10 ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin → (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) ∈ ω)
139129, 138syl 17 . . . . . . . . 9 (𝜑 → (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) ∈ ω)
140 f1fveq 7214 . . . . . . . . 9 ((𝐻:ω–1-1→𝑋 ∧ ((card‘𝑍) ∈ ω ∧ (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) ∈ ω)) → ((𝐻‘(card‘𝑍)) = (𝐻‘(card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))) ↔ (card‘𝑍) = (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))))
141135, 137, 139, 140syl12anc 835 . . . . . . . 8 (𝜑 → ((𝐻‘(card‘𝑍)) = (𝐻‘(card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))) ↔ (card‘𝑍) = (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))))
142133, 141mpbid 231 . . . . . . 7 (𝜑 → (card‘𝑍) = (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))
143142eqcomd 2737 . . . . . 6 (𝜑 → (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) = (card‘𝑍))
144 finnum 9893 . . . . . . . 8 ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ dom card)
145129, 144syl 17 . . . . . . 7 (𝜑 → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ dom card)
146 finnum 9893 . . . . . . . 8 (𝑍 ∈ Fin → 𝑍 ∈ dom card)
147116, 146syl 17 . . . . . . 7 (𝜑 → 𝑍 ∈ dom card)
148 carden2 9932 . . . . . . 7 (((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ dom card ∧ 𝑍 ∈ dom card) → ((card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) = (card‘𝑍) ↔ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
149145, 147, 148syl2anc 584 . . . . . 6 (𝜑 → ((card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) = (card‘𝑍) ↔ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
150143, 149mpbid 231 . . . . 5 (𝜑 → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍)
151 dfpss2 4050 . . . . . . . 8 ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 ↔ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍 ∧ ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍))
152151baib 536 . . . . . . 7 ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍 → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 ↔ ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍))
153128, 152syl 17 . . . . . 6 (𝜑 → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 ↔ ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍))
154 php3 9163 . . . . . . . . 9 ((𝑍 ∈ Fin ∧ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍) → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≺ 𝑍)
155 sdomnen 8928 . . . . . . . . 9 ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≺ 𝑍 → ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍)
156154, 155syl 17 . . . . . . . 8 ((𝑍 ∈ Fin ∧ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍) → ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍)
157156ex 413 . . . . . . 7 (𝑍 ∈ Fin → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 → ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
158116, 157syl 17 . . . . . 6 (𝜑 → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 → ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
159153, 158sylbird 259 . . . . 5 (𝜑 → (¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍 → ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
160150, 159mt4d 117 . . . 4 (𝜑 → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍)
161119, 160eleqtrrd 2835 . . 3 (𝜑 → (𝐻‘(card‘𝑍)) ∈ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))
162 fvex 6860 . . . 4 (𝐻‘(card‘𝑍)) ∈ V
163162eliniseg 6051 . . . 4 ((𝐻‘(card‘𝑍)) ∈ V → ((𝐻‘(card‘𝑍)) ∈ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ↔ (𝐻‘(card‘𝑍))(𝑊‘𝑍)(𝐻‘(card‘𝑍))))
164162, 163ax-mp 5 . . 3 ((𝐻‘(card‘𝑍)) ∈ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ↔ (𝐻‘(card‘𝑍))(𝑊‘𝑍)(𝐻‘(card‘𝑍)))
165161, 164sylib 217 . 2 (𝜑 → (𝐻‘(card‘𝑍))(𝑊‘𝑍)(𝐻‘(card‘𝑍)))
16626simprd 496 . . . . 5 (𝜑 → ((𝑊‘𝑍) We 𝑍 ∧ ∀𝑏 ∈ 𝑍 [(◡(𝑊‘𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊‘𝑍) ∩ (𝑣 × 𝑣))) = 𝑏))
167166simpld 495 . . . 4 (𝜑 → (𝑊‘𝑍) We 𝑍)
168 weso 5629 . . . 4 ((𝑊‘𝑍) We 𝑍 → (𝑊‘𝑍) Or 𝑍)
169167, 168syl 17 . . 3 (𝜑 → (𝑊‘𝑍) Or 𝑍)
170 sonr 5573 . . 3 (((𝑊‘𝑍) Or 𝑍 ∧ (𝐻‘(card‘𝑍)) ∈ 𝑍) → ¬ (𝐻‘(card‘𝑍))(𝑊‘𝑍)(𝐻‘(card‘𝑍)))
171169, 119, 170syl2anc 584 . 2 (𝜑 → ¬ (𝐻‘(card‘𝑍))(𝑊‘𝑍)(𝐻‘(card‘𝑍)))
172165, 171pm2.65i 193 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3060  {crab 3405  Vcvv 3446  [wsbc 3742   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913   ⊊ wpss 3914  ifcif 4491  𝒫 cpw 4565  {csn 4591  ∪ cuni 4870  ∩ cint 4912  ∪ ciun 4959   class class class wbr 5110  {copab 5172   Or wor 5549   We wwe 5592   × cxp 5636  ◡ccnv 5637  dom cdm 5638  ran crn 5639   “ cima 5641  Oncon0 6322  –1-1→wf1 6498  –1-1-onto→wf1o 6500  ‘cfv 6501  (class class class)co 7362   ∈ cmpo 7364  ωcom 7807   ↑m cmap 8772   ≈ cen 8887   ≼ cdom 8888   ≺ csdm 8889  Fincfn 8890  cardccrd 9880
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9586
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-oi 9455  df-card 9884
This theorem is referenced by:  pwfseqlem5  10608
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