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Theorem pwfseqlem4 9937
Description: Lemma for pwfseq 9939. Derive a final contradiction from the function 𝐹 in pwfseqlem3 9935. Applying fpwwe2 9918 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 2797 . . . . . . . . . . 11 𝑍 = 𝑍
2 eqid 2797 . . . . . . . . . . 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 8959 . . . . . . . . . . . . . 14 ω ∈ V
7 ovex 7055 . . . . . . . . . . . . . 14 (𝐴 ↑𝑚 𝑛) ∈ V
86, 7iunex 7532 . . . . . . . . . . . . 13 ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛) ∈ V
9 f1dmex 7521 . . . . . . . . . . . . 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 7352 . . . . . . . . . . . 12 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
1210, 11sylibr 235 . . . . . . . . . . 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 9936 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)
20 pwfseqlem4.z . . . . . . . . . . 11 𝑍 = ∪ dom 𝑊
214, 12, 19, 20fpwwe2 9918 . . . . . . . . . 10 (𝜑 → ((𝑍𝑊(𝑊‘𝑍) ∧ (𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍) ↔ (𝑍 = 𝑍 ∧ (𝑊‘𝑍) = (𝑊‘𝑍))))
223, 21mpbiri 259 . . . . . . . . 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 9907 . . . . . . . . . . . . 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 233 . . . . . . . . . . . 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 5126 . . . . . . . . 9 (𝜑 → 𝑍 ∈ V)
30 sseq1 3919 . . . . . . . . . . . . . 14 (𝑎 = 𝑍 → (𝑎 ⊆ 𝐴 ↔ 𝑍 ⊆ 𝐴))
31 id 22 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑍 → 𝑎 = 𝑍)
3231sqxpeqd 5482 . . . . . . . . . . . . . . 15 (𝑎 = 𝑍 → (𝑎 × 𝑎) = (𝑍 × 𝑍))
3332sseq2d 3926 . . . . . . . . . . . . . 14 (𝑎 = 𝑍 → ((𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ↔ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍)))
34 weeq2 5439 . . . . . . . . . . . . . 14 (𝑎 = 𝑍 → ((𝑊‘𝑍) We 𝑎 ↔ (𝑊‘𝑍) We 𝑍))
3530, 33, 343anbi123d 1428 . . . . . . . . . . . . 13 (𝑎 = 𝑍 → ((𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎) ↔ (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍)))
3635anbi2d 628 . . . . . . . . . . . 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 1111 . . . . . . . . . . . . . . 15 (((𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍)) ∧ (𝑊‘𝑍) We 𝑍) → (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍))
3938adantrr 713 . . . . . . . . . . . . . 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, 41syl6bbr 290 . . . . . . . . . . 11 (𝑎 = 𝑍 → ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)) ↔ 𝜑))
43 oveq1 7030 . . . . . . . . . . . . 13 (𝑎 = 𝑍 → (𝑎𝐹(𝑊‘𝑍)) = (𝑍𝐹(𝑊‘𝑍)))
4443, 31eleq12d 2879 . . . . . . . . . . . 12 (𝑎 = 𝑍 → ((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 ↔ (𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍))
45 breq1 4971 . . . . . . . . . . . 12 (𝑎 = 𝑍 → (𝑎 ≺ ω ↔ 𝑍 ≺ ω))
4644, 45imbi12d 346 . . . . . . . . . . 11 (𝑎 = 𝑍 → (((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 → 𝑎 ≺ ω) ↔ ((𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍 → 𝑍 ≺ ω)))
4742, 46imbi12d 346 . . . . . . . . . 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 6558 . . . . . . . . . . 11 (𝑊‘𝑍) ∈ V
49 sseq1 3919 . . . . . . . . . . . . . 14 (𝑠 = (𝑊‘𝑍) → (𝑠 ⊆ (𝑎 × 𝑎) ↔ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎)))
50 weeq1 5438 . . . . . . . . . . . . . 14 (𝑠 = (𝑊‘𝑍) → (𝑠 We 𝑎 ↔ (𝑊‘𝑍) We 𝑎))
5149, 503anbi23d 1431 . . . . . . . . . . . . 13 (𝑠 = (𝑊‘𝑍) → ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ↔ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)))
5251anbi2d 628 . . . . . . . . . . . 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 7031 . . . . . . . . . . . . . 14 (𝑠 = (𝑊‘𝑍) → (𝑎𝐹𝑠) = (𝑎𝐹(𝑊‘𝑍)))
5453eleq1d 2869 . . . . . . . . . . . . 13 (𝑠 = (𝑊‘𝑍) → ((𝑎𝐹𝑠) ∈ 𝑎 ↔ (𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎))
5554imbi1d 343 . . . . . . . . . . . 12 (𝑠 = (𝑊‘𝑍) → (((𝑎𝐹𝑠) ∈ 𝑎 → 𝑎 ≺ ω) ↔ ((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 → 𝑎 ≺ ω)))
5652, 55imbi12d 346 . . . . . . . . . . 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 8962 . . . . . . . . . . . . . . 15 ω ∈ On
58 onenon 9231 . . . . . . . . . . . . . . 15 (ω ∈ On → ω ∈ dom card)
5957, 58ax-mp 5 . . . . . . . . . . . . . 14 ω ∈ dom card
60 simpr3 1189 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑠 We 𝑎)
616019.8ad 2147 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ∃𝑠 𝑠 We 𝑎)
62 ween 9314 . . . . . . . . . . . . . . 15 (𝑎 ∈ dom card ↔ ∃𝑠 𝑠 We 𝑎)
6361, 62sylibr 235 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑎 ∈ dom card)
64 domtri2 9271 . . . . . . . . . . . . . 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 1896 . . . . . . . . . . . . . . . . 17 Ⅎ𝑟(𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))
67 nfcv 2951 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑟𝑎
68 nfmpo2 7100 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
6918, 68nfcxfr 2949 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑟𝐹
70 nfcv 2951 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑟𝑠
7167, 69, 70nfov 7053 . . . . . . . . . . . . . . . . . 18 Ⅎ𝑟(𝑎𝐹𝑠)
7271nfel1 2965 . . . . . . . . . . . . . . . . 17 Ⅎ𝑟(𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎)
7366, 72nfim 1882 . . . . . . . . . . . . . . . 16 Ⅎ𝑟((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎))
74 sseq1 3919 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = 𝑠 → (𝑟 ⊆ (𝑎 × 𝑎) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
75 weeq1 5438 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = 𝑠 → (𝑟 We 𝑎 ↔ 𝑠 We 𝑎))
7674, 753anbi23d 1431 . . . . . . . . . . . . . . . . . . 19 (𝑟 = 𝑠 → ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)))
7776anbi1d 629 . . . . . . . . . . . . . . . . . 18 (𝑟 = 𝑠 → (((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎) ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)))
7877anbi2d 628 . . . . . . . . . . . . . . . . 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 7031 . . . . . . . . . . . . . . . . . 18 (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
8079eleq1d 2869 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑠 → ((𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎) ↔ (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎)))
8178, 80imbi12d 346 . . . . . . . . . . . . . . . 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 1896 . . . . . . . . . . . . . . . . . 18 Ⅎ𝑥(𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))
83 nfcv 2951 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑥𝑎
84 nfmpo1 7099 . . . . . . . . . . . . . . . . . . . . 21 Ⅎ𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
8518, 84nfcxfr 2949 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑥𝐹
86 nfcv 2951 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑥𝑟
8783, 85, 86nfov 7053 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑥(𝑎𝐹𝑟)
8887nfel1 2965 . . . . . . . . . . . . . . . . . 18 Ⅎ𝑥(𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎)
8982, 88nfim 1882 . . . . . . . . . . . . . . . . 17 Ⅎ𝑥((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎))
90 sseq1 3919 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → (𝑥 ⊆ 𝐴 ↔ 𝑎 ⊆ 𝐴))
91 xpeq12 5475 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 = 𝑎 ∧ 𝑥 = 𝑎) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
9291anidms 567 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑎 → (𝑥 × 𝑥) = (𝑎 × 𝑎))
9392sseq2d 3926 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑎 × 𝑎)))
94 weeq2 5439 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → (𝑟 We 𝑥 ↔ 𝑟 We 𝑎))
9590, 93, 943anbi123d 1428 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎)))
96 breq2 4972 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → (ω ≼ 𝑥 ↔ ω ≼ 𝑎))
9795, 96anbi12d 630 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥) ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
9815, 97syl5bb 284 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝜓 ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
9998anbi2d 628 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))))
100 oveq1 7030 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
101 difeq2 4020 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑎))
102100, 101eleq12d 2879 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥) ↔ (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎)))
10399, 102imbi12d 346 . . . . . . . . . . . . . . . . 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 9935 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥))
10589, 103, 104chvar 2371 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎))
10673, 81, 105chvar 2371 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎))
107106eldifbd 3878 . . . . . . . . . . . . . 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 261 . . . . . . . . . . . 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 3505 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)) → ((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 → 𝑎 ≺ ω))
11247, 111vtoclg 3513 . . . . . . . . 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 8968 . . . . . . 7 (𝑍 ∈ Fin ↔ 𝑍 ≺ ω)
116114, 115sylibr 235 . . . . . 6 (𝜑 → 𝑍 ∈ Fin)
1175, 13, 14, 15, 16, 17, 18pwfseqlem2 9934 . . . . . 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 2886 . . . 4 (𝜑 → (𝐻‘(card‘𝑍)) ∈ 𝑍)
1204, 12, 24fpwwe2lem3 9908 . . . . . . . . . 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 683 . . . . . . . . 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 5832 . . . . . . . . . . . 12 (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ dom (𝑊‘𝑍)
12327simprd 496 . . . . . . . . . . . . . 14 (𝜑 → (𝑊‘𝑍) ⊆ (𝑍 × 𝑍))
124 dmss 5664 . . . . . . . . . . . . . 14 ((𝑊‘𝑍) ⊆ (𝑍 × 𝑍) → dom (𝑊‘𝑍) ⊆ dom (𝑍 × 𝑍))
125123, 124syl 17 . . . . . . . . . . . . 13 (𝜑 → dom (𝑊‘𝑍) ⊆ dom (𝑍 × 𝑍))
126 dmxpss 5911 . . . . . . . . . . . . 13 dom (𝑍 × 𝑍) ⊆ 𝑍
127125, 126syl6ss 3907 . . . . . . . . . . . 12 (𝜑 → dom (𝑊‘𝑍) ⊆ 𝑍)
128122, 127sstrid 3906 . . . . . . . . . . 11 (𝜑 → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍)
129116, 128ssfid 8594 . . . . . . . . . 10 (𝜑 → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin)
13048inex1 5119 . . . . . . . . . 10 ((𝑊‘𝑍) ∩ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) × (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))) ∈ V
1315, 13, 14, 15, 16, 17, 18pwfseqlem2 9934 . . . . . . . . . 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 2835 . . . . . . . 8 (𝜑 → (𝐻‘(card‘𝑍)) = (𝐻‘(card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))))
134 f1of1 6489 . . . . . . . . . 10 (𝐻:ω–1-1-onto→𝑋 → 𝐻:ω–1-1→𝑋)
13514, 134syl 17 . . . . . . . . 9 (𝜑 → 𝐻:ω–1-1→𝑋)
136 ficardom 9243 . . . . . . . . . 10 (𝑍 ∈ Fin → (card‘𝑍) ∈ ω)
137116, 136syl 17 . . . . . . . . 9 (𝜑 → (card‘𝑍) ∈ ω)
138 ficardom 9243 . . . . . . . . . 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 6892 . . . . . . . . 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 833 . . . . . . . 8 (𝜑 → ((𝐻‘(card‘𝑍)) = (𝐻‘(card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))) ↔ (card‘𝑍) = (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))))
142133, 141mpbid 233 . . . . . . 7 (𝜑 → (card‘𝑍) = (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))
143142eqcomd 2803 . . . . . 6 (𝜑 → (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) = (card‘𝑍))
144 finnum 9230 . . . . . . . 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 9230 . . . . . . . 8 (𝑍 ∈ Fin → 𝑍 ∈ dom card)
147116, 146syl 17 . . . . . . 7 (𝜑 → 𝑍 ∈ dom card)
148 carden2 9269 . . . . . . 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 233 . . . . 5 (𝜑 → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍)
151 dfpss2 3989 . . . . . . . 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 8557 . . . . . . . . 9 ((𝑍 ∈ Fin ∧ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍) → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≺ 𝑍)
155 sdomnen 8393 . . . . . . . . 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 261 . . . . 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 2888 . . 3 (𝜑 → (𝐻‘(card‘𝑍)) ∈ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))
162 fvex 6558 . . . 4 (𝐻‘(card‘𝑍)) ∈ V
163162eliniseg 5841 . . . 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 219 . 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 5441 . . . 4 ((𝑊‘𝑍) We 𝑍 → (𝑊‘𝑍) Or 𝑍)
169167, 168syl 17 . . 3 (𝜑 → (𝑊‘𝑍) Or 𝑍)
170 sonr 5391 . . 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 195 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1525  ∃wex 1765   ∈ wcel 2083  ∀wral 3107  {crab 3111  Vcvv 3440  [wsbc 3711   ∖ cdif 3862   ∩ cin 3864   ⊆ wss 3865   ⊊ wpss 3866  ifcif 4387  𝒫 cpw 4459  {csn 4478  ∪ cuni 4751  ∩ cint 4788  ∪ ciun 4831   class class class wbr 4968  {copab 5030   Or wor 5368   We wwe 5408   × cxp 5448  ◡ccnv 5449  dom cdm 5450  ran crn 5451   “ cima 5453  Oncon0 6073  –1-1→wf1 6229  –1-1-onto→wf1o 6231  ‘cfv 6232  (class class class)co 7023   ∈ cmpo 7025  ωcom 7443   ↑𝑚 cmap 8263   ≈ cen 8361   ≼ cdom 8362   ≺ csdm 8363  Fincfn 8364  cardccrd 9217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-inf2 8957
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-er 8146  df-map 8265  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-oi 8827  df-card 9221
This theorem is referenced by:  pwfseqlem5  9938
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