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Mirrors > Home > MPE Home > Th. List > infpssALT | Structured version Visualization version GIF version |
Description: Alternate proof of infpss 9628, shorter but requiring Replacement (ax-rep 5154). (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
infpssALT | ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ominf4 9723 | . 2 ⊢ ¬ ω ∈ FinIV | |
2 | reldom 8498 | . . . . 5 ⊢ Rel ≼ | |
3 | 2 | brrelex2i 5573 | . . . 4 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
4 | isfin4 9708 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ FinIV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴))) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (ω ≼ 𝐴 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴))) |
6 | domfin4 9722 | . . . 4 ⊢ ((𝐴 ∈ FinIV ∧ ω ≼ 𝐴) → ω ∈ FinIV) | |
7 | 6 | expcom 417 | . . 3 ⊢ (ω ≼ 𝐴 → (𝐴 ∈ FinIV → ω ∈ FinIV)) |
8 | 5, 7 | sylbird 263 | . 2 ⊢ (ω ≼ 𝐴 → (¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) → ω ∈ FinIV)) |
9 | 1, 8 | mt3i 151 | 1 ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 ⊊ wpss 3882 class class class wbr 5030 ωcom 7560 ≈ cen 8489 ≼ cdom 8490 FinIVcfin4 9691 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-om 7561 df-er 8272 df-en 8493 df-dom 8494 df-fin4 9698 |
This theorem is referenced by: (None) |
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