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Theorem infpssALT 10398

Description: Alternate proof of infpss 10301, shorter but requiring Replacement (ax-rep 5231). (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

**Assertion**
Ref	Expression
infpssALT	⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴))

Distinct variable group: 𝑥,𝐴

**Proof of Theorem infpssALT**
Step	Hyp	Ref	Expression
1		ominf4 10397	. 2 ⊢ ¬ ω ∈ Fin^IV
2		reldom 8987	. . . . 5 ⊢ Rel ≼
3	2	brrelex2i 5708	. . . 4 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
4		isfin4 10382	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ Fin^IV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)))
5	3, 4	syl 18	. . 3 ⊢ (ω ≼ 𝐴 → (𝐴 ∈ Fin^IV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)))
6		domfin4 10396	. . . 4 ⊢ ((𝐴 ∈ Fin^IV ∧ ω ≼ 𝐴) → ω ∈ Fin^IV)
7	6	expcom 419	. . 3 ⊢ (ω ≼ 𝐴 → (𝐴 ∈ Fin^IV → ω ∈ Fin^IV))
8	5, 7	sylbird 263	. 2 ⊢ (ω ≼ 𝐴 → (¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) → ω ∈ Fin^IV))
9	1, 8	mt3i 150	1 ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∃wex 1811 ∈ wcel 2144 Vcvv 3450 ⊊ wpss 4012 class class class wbr 5102 ωcom 7881 ≈ cen 8978 ≼ cdom 8979 Fin^IVcfin4 10365

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7882 df-er 8725 df-en 8982 df-dom 8983 df-fin4 10372

This theorem is used by: (None)

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